Modeling and Simulation of the Heat Transfer Behaviour of a Shell-and-Tube Condenser for a Moderately High-Temperature Heat Pump 79 The estimation procedures for sizing a shell-and-tube condenser is shown as follows: • Input design parameters: • Input design parameters include: refrigerant inlet/outlet temperatures, refrigerant inlet pressure, water inlet/outlet temperatures, water and refrigerant mass flow rates, condensing temperature, number of copper tubes, tube inner/outer diameters, shell inner diameter, baffle spacing, and copper tube spacing • Give a tube length and shell-side outlet temperature to be initial guess values for Section-I calculation • Calculate the physical properties for Section-I and Section-II • Calculate the overall heat transfer rates by present model • Check the percent error between model predicting and experimental data for overall heat transfer rates If the percent error is less than the value of 0.01%, then output the tube length and end the estimation process; if it is larger than the percent error, then set a new value for L and return to the second step In accordance with the above estimation procedures, the resulting length is 0.694 m when input the experimental data set, Case 1, as the design parameters for sizing The same estimation procedures are utilizing to another 26 cases, and the results are shown in Figure 10 Fig 10 Estimation results for sizing condensers Comparisons between the estimating values length for all the cases and the experimental data (0.7 m) indicats that the relative error were within ± 10 % with an average CV value of 3.16 % In summary, the results from the application of present model on heat exchanger sizing calculation are satisfactory 5.2 Rating problem (Estimation of thermal performance) For performance rating procedure, all the geometrical parameters must be determined as the input into the heat transfer correlations When the condenser is available, then all the geometrical parameters are also known In the rating process, the basic calculation is the 80 Two Phase Flow, Phase Change and Numerical Modeling calculations of heat transfer coefficient for both shell- and refrigerant-side stream If the condenser's refrigerant inlet temperature and pressure, water inlet temperature, hot water and refrigerant mass flow rates, and tube size are specified, then the condenser's water outlet temperature, refrigerant outlet temperature, and heat transfer rate can be estimated The estimation process for rating a condenser: • Input design parameters: The input design parameters include: refrigerant inlet/outlet temperatures, refrigerant inlet pressure, water inlet temperature, mass flow rate of hot water/refrigerant, and geometric conditions • Give a refrigerant outlet temperature as an initial guess for computing the hot water  Qw outlet temperature: Two = Twi +  mwC pw • • • • Give an outlet temperature (Tr) as an initial guess for Section-I Calculate the properties for Section-I and Section-II Calculate the overall heat transfer rates by present model Check the percent error between model predicting and experimental data for overall heat transfer rates If the percent error is less than the value of 0.01%, then output the refrigerant outlet temperature, water outlet temperature, and heat transfer rate; if it is larger than the percent error, then reset a new refrigerant outlet temperature, and return to the second step In accordance with the above calculation process, the experimental data of Case 1 can be used as input into the present model for rating calculations The calculation results give the water outlet temperature is 74.84°C, refrigerant outlet temperature is 64.35°C, and heat transfer rate was is 33.01 kW Experimental data of Case 2 were used as input into the rating calculation process, and another set of result tell: water outlet water temperature is 45.16 °C, refrigerant outlet temperature is 39.04 °C, and heat transfer rate is 35.03 kW Repeat the same procedures for the remaining 26 sets of experimental data, the calculation results for rating are displayed in Figures 11 As depicted in Figure 11, comparison of the model predicting and the experimental data for water outlet temperature, refrigerant outlet temperature and heat transfer rates show that the average CV values are 0.63%, 0.36%, and 1.02% respectively In summary, the predicting accuracies of present model on shell-and-tube condenser have satisfactory results 6 Conclusion This study investigated the modelling and simulation of thermal performance for a shelland-tube condenser with longitude baffles, designed for a moderately high-temperature heat pump Through the validation of experimental data, a heat transfer model for predicting heat transfer rate of condenser was developed, and then used to carry out size estimation and performance rating of the shell-and-tube condenser for cases study In summary, the following conclusions were obtained: • A model for calculation, size estimation, and performance rating of the shell-and-tube condenser has been developed, varified, and modified A good agreement is observed between the computed values and the experimental data • In applying the present model, the average deviations (CV) is within 3.16% for size estimation, and is within 1.02% for performance rating Modeling and Simulation of the Heat Transfer Behaviour of a Shell-and-Tube Condenser for a Moderately High-Temperature Heat Pump Fig 11 Simulation results of rating condensers for (a) water outlet temperature, (b) refrigerant outlet temperature, and (c) heat transfer rate 81 82 Two Phase Flow, Phase Change and Numerical Modeling 7 References Allen, B., & Gosselin, L (2008) Optimal geometry and flow arrangement for minimizing the cost of shell-and-tube condensers International Journal of Energy Research, Vol 32, pp 958-969 Caputo, A.C., Pelagagge, P.M., & Salini, P (2008) Heat exchanger design based on economic optimisation Applied Thermal Engineering,Vol 28, 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algorithms from economic point of view Chemical Engineering and Processing, Vol 45, pp 268-275 Vera-García, F., García-Cascales, J.R., Gonzálvez-Maciá, J., Cabello, R., Llopis, R., Sanchez, D., ＆ Torrella, E (2010) A simplified model for shell-and-tubes heat exchangers: Practical application Applied Thermal Engineering, Vol 30, pp 1231-1241 Wang, Q.W., Chen, G.D., Xu, J., & Ji, Y.P (2010) Second-Law Thermodynamic Comparison and Maximal Velocity Ratio Design of Shell-and-Tube Heat Exchangers with Continuous Helical Baffles Journal of Heat Transfer, Vol 132, pp 1-9 4 Simulation of Rarefied Gas Between Coaxial Circular Cylinders by DSMC Method H Ghezel Sofloo Department of Aerospace Engineering, K.N.Toosi University of Technology, Tehran Iran 1 Introduction In every system, if Knudsen number is larger than 0.1, the Navier-Stokes equation will not be satisfied for investigation of flow patterns In this condition, the Boltzmann equation, presented by Ludwig Boltzmann in 1872, can be useful The conditions that this equation can be used were investigted by Cercignani in 1969 The most successful method for solving Boltzmann equation for a rarefied gas system is Direct Simulation Monte Carlo (DSMC) method This method was suggested by Bird in 1974 The cylindrical Couette flow and occurrence of secondry flow (Taylor vortex flow) in a annular domin of two coaxial rotating cylinders is a classical problem in fluid mechanics Because this type of gas flow can occur in many industrical types of equipment used in chemical industries, Chemical engineers are interested in this problem In 2000, De and Marino studied the effect of Knudsen number on flow patterns and in 2006 the effect of temperature gradient between two cylinders was investigated by Yoshio and his co-workers The aim of the present paper is investigation of understanding of the effect of different conditions of rotation of the cylinders on the vortex flow and flow patterns 2 Mathematical model In the Boltzmann equation, the independent variable is the proption of molecules that are in a specific situation and dependent variables are time, velocity components and molecules positions We consider the Boltzmann equation as follow:  δf  δf  δf 1 + ν  + F  = Q( f , f ) δt δx δν Kn (1) The bilinear collision operator, Q(f,f), describes the binary collosion of the particles and is given by: Q( f , f ) =    σ ( ν −ν * , ω )( f ′ f *′ − ff * )dω dν * (2) R3 S 2 Where, w is a unit vector of the shere S2, so w is an element of the area of the surface of the unit sphere S2 in R3 With using this assumption that f is zero, we can rewrite equation 2 as: 84 Two Phase Flow, Phase Change and Numerical Modeling Q( f , f ) =     σ ( ν −ν *  , ω ) f ′ f *′dω dν * − R3 S2     σ ( ν −ν  , ω ) ff * dω dν * * R3 S2 (3) The sign ‘ is refered to values of distribution function after collision The value of above  integral is not related on V , then we have Q( f , f ) =     σ ( ν −ν *  , ω ) f ′ f *′dω dν * − f R3 S 2     σ ( ν −ν *  , ω ) f * dω dν * R3 S2 (4) Inasmuch as the values of distribution function depend on its value before collision, we have:  Q( f , f ) = P( f , f ) − f μ (ν ) (5)   Where μ (ν ) is the mean value of the collision of the particles that move with ν velocity Then we can estimate μ (ν ) as  μ (ν ) = μ = κρ m (6) Then the Boltzmann equation can be written as  δf  δf 1  P( f , f ) − f μ (ν ) +ν  =   δt δ x Kn (7) For solving this equation, we use fractional step method, so we have  δf δf = −ν  δt δx (8)  δf 1 1  P( f , f ) − f μ (ν ) = Q( f , f ) =  δ t Kn Kn  (9) Equation 8 describes the movement of the particles and equation 9 explains the collision of the particles For estimation of new position of a mobile particle, we use following realationship  new  old  (10) x = x + ν Δt For solving equation 9 by a numerical method, we can write it as f n+1 − f n 1  P( f n , f n ) − μ f n  =  Kn  Δt (11) If we rearrange it, we will have f n+1 = μΔt P( f n , f n ) μΔt n )f + (1 − Kn Kn μ (12) Simulation of Rarefied Gas Between Coaxial Circular Cylinders by DSMC Method 85 The first term on the right side of Eq (12) is refered to probability of collision and the second term is refred to situation that no collision occurs Equation (12) is solved using the DSMC method DSMC is a molecule-based statistical simulation method for rarefied gas introduced by Bird (2) It is a numerical solution method to solve the dynamic equation for gas flow by at least thousands of simulated molecules Under the assumption of molecular chaos and gas rarefaction, the binary collisions are only considered Therefore, the molecules' motion and their collisions are uncoupling if the computational time step is smaller than the physical collision time After some steps, the macroscopic flow characteristics should be obtained statistically by sampling molecular properties in each cell and mean value of each property should be recorded For estimation of macroscopic characteristics we used following realationship ρ=   fdν (13) ℜ3    ρ u =  ν fdν (14)  1    3 (ν − u) fdν 3ρ ℜ (15) ℜ3 T= p= ρ m kT (16) 3 Results and discussion We consider a rarefied gas inside an annular domin of coaxial rotating cylinders The radius of the inlet and the outlet cylinder are R1 and R2 (R1