Mathematical Modelling for Thermal and Mechanical Design of Shell and Tube Type Gas Cooler Used in Transcritical CO2 Refrigeration System

This paper is about Mathematical Modelling for Thermal and Mechanical Design of Shell and Tube type Gas Cooler used in Transcritical CO2 Refrigeration system. Transcritical refrigeration system refers to system whose condenser temperature is above critical temperature of refrigerant. To achieve it, the condenser in conventional refrigeration system is replaced by Gas Cooler where Refrigerant vapour is cooled sensibly without condensation. The gas cooler is used for cooling of refrigerant by using water as coolant. The temperature of refrigerant vapour coming out of compressor in transcritical system is more as compared to conventional refrigerant system. So gas cooler can be effectively used for heating of water. This paper describes a mathematical model that can be used in predicting the heat transfer performance of a shell and tube type Gas Cooler used in transcritical CO2 refrigeration system. The model uses Kern Method of Heat exchanger design. Given the fluid inlet and outlet temperatures flow rates of fluid & fluid properties the model determines (a) the necessary heat transfer surface area, (b) Outside and inside heat transfer coefficient, (c) overall heat transfer coefficient, (d) Pressure drop on shell and tube side, (e) It also determines mechanical design parameters such as shell O. D, Shell thickness, Tube sheet thickness, Flange thickness.


Introduction
In the last few years researchers came to know about the adverse effect of refrigerants on environment i.e. global warming and ozone depletion potential as these are adverse effects government added restrictions on use of such refrigerant who causes the global warming and ozone depletion, so industries are in search of new and natural refrigerant which has low global warming and less ozone depletion potential. In this regard, CO 2 is one of the refrigerants, which has zero ozone depletion potential and less value of global warming potential. CO 2 is having less value of critical temperature and pressure so getting less temperature range for application and restricted to 31°C to use CO 2 for wide range of temperature to operate the system at pressure above critical point pressure i.e. above 73 bar as the pressure of the system is above critical point the condensation is not possible at pressure above critical pressure so in CO 2 transcritical Refrigeration system condenser is replaced by gas cooler. In this paper Shell and Tube type Gas Cooler used in transcritical CO 2 refrigeration system is modelled using kern method as shown in figure 1. The various mathematical model used for design of shell and tube type gas cooler is explained in below section.

Method of Design
(1) Formulation of mathematical models for thermal and mechanical Design (2) Calculation of different design parameters (3) Selection of required parameters from standard charts and tables (4) Final design of shell and tube gas cooler

Heat Duty of Heat Exchanger (Q)
In the shell-and-tube condenser the water flows outside the tubes and refrigerant flows inside the tubes through the shell.
Heat duty of Heat Exchanger (Q) by performing Energy Balance Heat Duty (Q) = ṁ w x C p x (T w2 -T w1 ) By considering uncertainty in heat load multiply Heat Duty by 1.25 to compensate Uncertainty of Heat Duty Due change inlet temperature of Cold fluid or Hot Fluid Corrected Heat Duty (Q cor ) = 1.25 x Q Where ṁ w is mass flow rate of water (kg/s) and C p is Specific Heat of water (J/kg°C), T w1 and T w2 are respectively inlet and outlet temperature of water.

Log-Mean Temperature Difference (LMTD)
The log mean temperature difference ∆T m for counter current flow is determined by: Where ∆T 1 = (T c1 -T w2 ) & ∆T 2 = (T c2 -T w1 ), T w1 and T w2 are respectively the inlet and outlet temperature for water, Tc 1 and Tc 2 are the inlet and outlet temperature for CO 2 respectively

Correction Factor
In design the heat exchangers, a correction factor is applied to the log mean temperature difference (LMTD) to allow for the departure from true counter current flow to determine the true temperature difference. Correction factor for LMTD can be calculated from the Standard Graph The value of correction factor obtained from graph using R= 0.711 S=0.789 and 2 shell passes and 4 tube passes is 0.78

Heat Transfer Area
The heat transfer area (A) of the shell-and-tube condenser is computed by: Where Ft is LMTD correction factor and Uss is assumed overall heat transfer coefficient (w/m 2 k).

Number of Tubes (Nt)
The Number of tubes required for required surface area is given by: Where D o is outside diameter of tube (meter) and L is length of tube (meter).

Inside Heat Transfer Coefficient
Inside Heat transfer Coefficient Using sieder-Tate equation is given by: Where h i is inside heat transfer coefficient (w/m 2 k), (µ w ) is Dynamic viscosity at wall temperature (N. S/m 2 ), µ is Dynamic viscosity at bulk mean temperature (N. S/m 2 ), C pc is Specific Heat of CO 2 (J/kg⁰K), Kc is Thermal Conductivity of CO 2 (W/m⁰K) and Di is inside diameter of tube.

Shell Side Heat Transfer Coefficient h o
For calculation of h o first one has to calculate Re for Shell side fluid for that purpose one must know the Mass velocity (Gs) of shell side fluid, Cross flow area (a c ).
For calculating Mass velocity equivalent diameter of Shell (De) is required. Where Kw is thermal conductivity of water (w/m°C), h o is outside heat transfer coefficient, G s is Mass flow Velocity (m/sec), a c is cross flow area, Pt is pitch of tube (meter), B is baffle spacing (meter), Ds is shell inside diameter (meter), ρ w is density of water (kg/m 3 ) and ν w is Kinematic Viscosity (m 2 /s).

Overall Heat Transfer Coefficient
Overall heat transfer coefficient U based on inside surface area depends on the tube inside diameter, tube outside diameter, tube side convective coefficient, shell side convective coefficient, tube side fouling resistance, shell side fouling resistance, and tube material is given by:

Shell Thickness (t s )
The shell thickness ( ) can be calculated from the equation below based on the maximum allowable stress and corrected for joint efficiency Where P is Design pressure (N/mm 2 ), f is Maximum allowable stress of the material used for construction (N/mm 2 ), J is Joint efficiency (usually varies from 0.7 to 0.9) and C a corrosion allowance.

Tube Sheet Thickness
The minimum tube-sheet thickness (TEMA standard) to 'resist bending' can be calculated by, Where f is for fixed tube sheet, G p is diameter over which pressure is acting ( for fixed tube sheet it is equal to shell inside diameter ), P is shell pressure (N/mm 2 ) and K is Mean ligament efficiency.
For square pitch K is given by: For 19 mm outside diameter of tube the minimum tube sheet thickness is 15 mm, as calculated tube sheet thickness Used in Transcritical CO 2 Refrigeration System is greater than 15 mm design is satisfied.
Gasket Design A preliminary estimation of gaskets is done using following expression: Residual gasket force=Gasket seating force-(Hydrostatic pressure force) The residual gasket force should be greater than that required to prevent the leakage of the internal fluid. This condition results the final expression in the form of Where m is gasket factor for Flat iron jacketed asbestos (m

Results
By using the mathematical models mentioned above the different parameters are calculated and tabulated in subsequent tables. For calculation of parameters some parameters are directly taken from standard tables and standard graphs. Mass flow rate of water(ṁw) 0.0389kg/s

Conclusion
When we compare the results of shell & Tube heat exchanger used in conventional refrigeration system & transcritical CO 2 refrigeration system we found out that: (1) Design pressure is higher in Gas Cooler so the thickness of shell, Tubes, flanges, tube sheet are higher than conventional shell and tube heat exchanger. (2) Due to high pressure in CO 2 refrigeration system the material used for construction of tube is stainless steel rather than copper and brass. (3) The CO 2 gas cooler is working under higher pressure than conventional shell and tube heat exchanger. (4) As temperature of refrigerant vapour is higher than conventional refrigeration system gas cooler can be effectively used for water heating for domestic & industrial purpose.