Thermal design calculation method of the whole coil closed cooling tower with condensation inside the tube

In this paper, the heat exchange process of a closed cooling tower with condensation inside the tube is analyzed. Combined with the calculation method of a traditional closed cooling tower, the thermal model and mathematical model of the whole coil closed cooling tower with condensation inside the tube are established. The fourth-order Runge-Kutta (RK4) method is used to write a program to solve the differential equation. A case is calculated to solve the heat exchange area and spray water temperature. Its feasibility for design reference is verified.


Introduction
A cooling tower is a common thermal equipment for cooling circulating water, which is widely used in the field of energy and chemical industry.The thermal research of the cooling tower is also very extensive at home and abroad.Hassab et al. [1] studied the cross-flow closed cooling tower (CCCT) by experimental method and numerical simulation and developed the functionality and simplified correlation of CCCT effectiveness.Fang et al. [2] analyzed the influencing factors of the closed wet cooling tower (CWCT) and fitted the experimental data of this cooling tower.A new formula for calculating the heat and mass transfer coefficient was obtained, which can be used to guide the efficiency of the cooling tower.Zhou et al. [3] analyzed the extent to which packing and cooling water flow direction affect the thermal performance of cooling towers and verified the validity of their proposal to predict effluent temperatures.Xie et al. [4] introduced a model aimed at analyzing heat and mass transfer occurring at the interface among multiple tubes.Their findings suggest that the influence of air Reynolds number on the heat transfer efficiency of a CWCT outweighs the influence of shower water Reynolds number on the same efficiency.Mahdi and Jaffal [5] carried out experiments focusing on various parameters of cooling towers.They investigated how the positioning of packing material affects the heat transfer performance of the closed cooling tower (CCT) and made predictions regarding heat and mass transfer coefficients.
There are few studies on the CCT with condensation in the tube, especially for the design and calculation of this tower type, with a lack of corresponding literature reference.In this paper, the whole coil-closed cooling tower with condensation in the tube is analyzed and studied, and its thermal model is established.The calculation method is provided and used for design and calculation.
For the design of a CCT, the design parameters are usually given.The design results are not only related to the water spray density and cross-section wind speed in the selected tower but also to the IOP Publishing doi:10.1088/1742-6596/2771/1/012021 2 material and size of the selected heat exchanger coil and the internal and external shape of the tube, the configuration of the tube bundle and the interaction between the flow direction of the cooling fluid and the incoming air and spray water from the external environment.The presence and properties of packing material in the tower are interconnected with these factors.The thermal calculation we do should take into account the cooling tower area, variable condition characteristics, energy consumption, water consumption, but also consider the stability of the whole cooling tower and the convenience of maintenance.Referring to the calculation method of the closed cooling tower [6][7][8], the design and calculation of the whole coil CCT with condensation in the tube are carried out.

Mathematical model
The initial design conditions of the CCT in this calculation book are: atmospheric pressure pa, the flow rate Mf of the cooled fluid, the inlet temperature tfi of the cooled fluid, the outlet temperature tfo of the cooled fluid, the dry bulb temperature ș of the air entering the tower, and the wet bulb temperature Ĳ of the air entering the tower.
When the fluid in the CCT flows through the coil, the temperature distribution of the fluid and spray water in the CCT section is shown in Figure 1.The temperature distribution of the fluid, spray water, and air in the tube along the direction of the fluid flow in the tube is shown in Figure 2.Over the micro heat transfer area ‫ݔ݀‬ along the entire coil of a CCT, heat is transferred from the fluid inside the tube to the surroundings, resulting in a reduction in the temperature of the fluid inside the tube: --The heat transfer coefficient from the fluid in the tube to the spray cooling water.During the condensation heat transfer process within the tube, the temperature ‫ݐ‬ should remain at its saturation temperature ‫ݐ‬ ௦ .Consequently, heat transfer from the fluid inside the tube leads to a decrease in its dryness.Equations ( 1a) and (1b) should be written as in the equation: ߮--The dryness of the fluid in the tube.ߛ --Latent heat of vaporization of fluid in tube.
The transfer of heat from the fluid inside the pipe to the spray water, and subsequently from the spray water to the air, collectively contribute to the change of the spray water temperature: in the equation: ‫ܯ‬ ௪ --The flow of spray water.ܿ ௪ --The specific heat of spray water;(J/kg•ႏ) ‫ܭ‬ ெ --The mass transfer coefficient of spray water to airflow, where ǻX refers to the difference of moisture content.
݅ ‫כ‬ --The enthalpy of saturated moist air corresponding to the temperature of spray water.It is a function of spray water temperature: The heat transferred from the spray water to the air results in an increase in the enthalpy of the air: in the equation: ‫ܯ‬ --Dry air mass flow in moist air; When the fluid in the tube is single-phase flow heat transfer, the ordinary differential equations of the parameters of fluid temperature ‫ݐ‬ , spray water temperature ‫ݐ‬ ௪ and air enthalpy ݅ in the CCT with respect to the heat transfer area ‫ݔ‬ can be obtained by combining Equations (1b), (2b) and (3b).
When the fluid in the tube is condensed convective heat transfer, the ordinary differential equations of the parameters of fluid dryness ɔ, spray water temperature‫ݐ‬ ௪ and air enthalpy ݅ in the CCT with respect to the heat transfer area ‫ݔ‬ can be obtained by combining Equations (1d), (2b) and (3b).
The boundary conditions corresponding to Equation ( 4), or Equation ( 5) are: The meaning of the last equation in the boundary condition is that the spray water is reused, so the temperature ‫ݐ‬ ௪ of the spray water at the beginning of the coil is equal to the temperature ‫ݐ‬ ௪ at the end of the coil.

Calculation of heat transfer coefficient ‫ܭ‬
In the above calculation model, the calculation method of ‫ܭ‬ involved is described in detail below.
in the equation: ߙ --The convective heat transfer coefficient between the fluid in the tube and the inner surface of the tube.
ߙ --The convective heat transfer coefficient between the spray water outside the tube and the outer surface of the tube.
‫ݎ‬ , ‫ݎ‬ --It is the fouling thermal resistance on the inner and outer walls of the tube respectively.;[(m 2 •ႏ)]/W] ݀ , ݀ --They are inner and outer diameters respectively;(m) (1) Calculation of convective heat transfer coefficient ߙ in tube If the fluid in the tube is single-phase flow heat transfer, the convective heat transfer coefficient between the fluid in the tube and the inner surface of the tube ߙ in the form of the Gnielinski associative equation: If the fluid in the tube is single-phase flow heat transfer, the convective heat transfer coefficient ߙ between the fluid and the inner surface of the tube the following equation is used to calculate: --The fluid gas phase friction multiplier in the tube is defined as follows: ܺ ௧௧ --The Lockhart-Martinelli parameter is defined as follows: ‫ܬ‬ ௩ --The dimensionless gas phase velocity of the fluid in the tube is defined as follows: Fr ௩ --The gas-phase Froude number of the fluid in the tube is defined as follows: IOP Publishing doi:10.1088/1742-6596/2771/1/012021 (2) Calculation of convective heat transfer coefficient ߙ in tube ‫ܣ‬ --Minimum circulation cross-sectional area, [m 2 ], ‫ܣ‬ = ݊ ‫ݏ(‬ െ ݀ ) × ‫ܮ‬ , where ݊ , ‫ݏ‬ , ݀ , ‫ܮ‬ is the number of columns of the heat exchanger (the number of tubes in a row), gauge, tube diameter, and tube length.

Calculation of mass transfer coefficient ‫ܭ‬ ெ
The calculation method of mass transfer coefficient ‫ܭ‬ ெ is described in detail as follows.
in the equation: ‫ܭ‬ --The mass transfer coefficient between spray water and airflow: ݉ --The slope m of the saturation curve in the temperature-humidity diagram of moist air (value at spray water temperature); ߙ ᇱ --Convective heat transfer coefficient between spray water and air, water interface.In the range of 1.389 < ௰ ௗ < 3.056, 0.694 < M ୫ୟ୶ < 5.278: IOP Publishing doi:10.1088/1742-6596/2771/1/0120217

Solution of the mathematical model
The thermodynamic design and calculation of the CCT with condensation in the tube, the differential Equation group ( 4) is required to be solved in the heat transfer section of the single relative flow of the fluid in the tube, and the differential Equation group ( 5) is required to be solved in the condensation section of the fluid in the tube.The boundary condition of the differential equation is Equation (6).
Because the calculation of the parameters involved in the differential equation is very complicated, the differential equation group can not obtain the analytical solution and can only be solved by numerical method.This calculation manual uses the RK4 method to write programs to solve the differential equations described above.The method divides the entire cooling tower heat transfer region into segments, starting at the outlet end of the fluid in the tubes, and calculates the heat transfer and each of the unmeasured quantities ‫ݐ‬ (or ߮), ‫ݐ‬ ௪ , and ݅, segment by segment.
The fourth-order Runge-Kutta method is briefly described below.Equations ( 4) or ( 5) can be written as: in the equation: ‫ݕ‬ ଵ , ‫ݕ‬ ଶ , … … , ‫ݕ‬ is each unknown quantity.
If the unknown quantity ‫ݕ‬ റ at one end of a certain section, the unknown quantity ‫ݕ‬ റ ାଵ at the other end of this section can be calculated according to the following equation: Among them, ‫ܨ݀‬ is the heat transfer area of this small section ݇ ሬ ሬറ 1 , ݇ ሬ ሬറ 2 , ݇ ሬ ሬറ 3 and ݇ ሬ ሬറ 4 are the calculated intermediate rate of change of each variable, calculated according to the following equation:

Calculation flow
Using the above algorithm to calculate the variation of each unmeasured ‫ݐ‬ ݂ (or ߮), ‫ݐ‬ ‫ݓ‬ and ݅ with the heat transfer area, it is essential to give the area ‫ܨ‬ and the temperature of the spray water at the air inlet ‫ݐ‬ ௪ in advance.These two parameters are the design results of the CCT.It is necessary to design a certain calculation process to make the assumed area ‫ܨ‬ meet the requirements of heat transfer, and the spray water temperature is equal at both ends of the coil ‫ݐ‬ ௪ = ‫ݐ‬ ௪ .The design calculation process used in this calculation book is shown in Figure 3.
In the calculation process, the Newton iteration method is used to accelerate the convergence when repeated assumptions are necessary for the spray water temperature ‫ݐ(‬ ௪ ) and the area F.

Physical parameters
The thermophysical parameters of all heat transfer fluids involved in this design calculation book are calculated by the sixth-order fitting polynomial of the corresponding temperature, namely: The data of polynomial fitting are the data in the range of 0~100°C obtained by REFPROP software.The following Table 1 shows the polynomial coefficients for the calculation of the physical parameters of methanol.The design conditions are input into the written program, and the final drawing result is shown in Figure 4: Finally, the heat exchange area is 81.32 m², and the spray water temperature is 32.28°C.

Conclusion
Derived from the principle of heat and mass transfer and the differential equation of a closed cooling tower with condensation in a tube, the mathematical model is established.The calculation program can accurately and quickly solve the heat transfer area required by the cooling tower, which is greatly convenient for engineering requirements.At the same time, considering the test and experience of coil manufacturers, it provides a theoretical support for the selection and design of the CCT.

Figure 1 .
Figure 1.Fluid temperature distribution on the cross section of CCT.

Figure 2 .
Figure 2. Temperature distribution of the fluid along the direction of fluid flow in the tube.
number, ߣ is the thermal conductivity of the fluid in the tube; [W/(m•ႏ)]; number, ߥ is the kinematic viscosity of the fluid in the tube;(m 2 /s), ‫ݑ‬ is the velocity of the fluid in the tube;(m/s) number, ߤ is the dynamic viscosity of the fluid in the tube, ܿ is the constant pressure specific heat capacity of the fluid in the tube; (J/kg•ႏ) ݂ --It is the turbulent flow resistance coefficient of the fluid in the tube, its calculation equation is: ݂ = (1.8lgRe െ 1.5) ିଶ flux density of fluid in the tube; [kg/(m 2 •s)] Pr --Prandtl number of liquid phases in tube ߤ --Dynamic viscosity of liquid phase in tube;(Pa•s) ߤ ௩ --Dynamic viscosity of fluid gas phase in tube;(Pa•s) ߩ --The density of liquid phase in the tube;(kg/m 3 ) ߩ ௩ --The density of fluid phase in the tube;(kg/m 3 ) ߣ --Thermal conductivity of liquid phase in tube; [W/(m•ႏ)] ߖ ௩

Figure 4 .
Figure 4. Result drawing.Finally, the heat exchange area is 81.32 m², and the spray water temperature is 32.28°C.

Table 1 .
Calculation coefficient table of methanol physical parameters.coilCCT with condensation inside the coil is designed.The circulating working fluid in the coil is processed fluid R134 A, and the inlet is saturated liquid.The specific design conditions are shown in Table2.