Mathematical Model of Heat Pumps' Coaxial Evaporator with Distributed Parameters

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1 Mathematical Model o Heat Pumps' Coaxial Evaporator with Distributed Parameters Jozse Nyers Obuda University Budapest, Hungary Subotica Tech, Marko Oreskovic 16, Subotica, Serbia nyers@uni-obuda.hu Zoltan Pek Obuda University Budapest, Hungary Ivan Saric, High School, Subotica, Serbia messc@tehnickaskolasubotica.edu.rs Abstract: The aim o this article is to create a very precise mathematical model with distributed parameters o the heat pumps' coaxial evaporator in order to examine the behavior o evaporator in the steady-state operation mode. The mathematical model consists o the balance equations, the equations o viscous riction coeicient, the convective heat transer, and the state equation o rerigerant R134a. The boundary conditions, which are the thermal-expansion valve and the compressor, in this work, have not yet been taken into account. It is the task and the goal o the next article. For the solution o mathematical model an iterative numerical procedure was applied. Simulation results were obtained in numerical orm and are presented graphically. The new in this paper: implanted the Kandlikar's evaporative heat transer coeicient as a unction o vapor quality, applied the viscous riction coeicient o the rerigerant according to Friedel and state equations o R134a according to Martin-Hou. In addition, using the new mathematical model simulated the behavior o coaxial evaporator. Keywords: coaxial evaporator; heat pump; perormance; mathematical model; distributed parameters; eiciency Nomenclature m Mass low rate [kg/s] α Convective heat transer coeicient [W/m 2 / ] λ C p t, T Conductive heat transer coeicient [W/m/ ] Speciic heat, p = const [J/kg/ ] Temperature [ ], [K] 41

2 J. Nyers et al. Mathematical Model o Heat Pumps' Coaxial Evaporator with Distributed Parameters dt Temperature dierence [ ] Function o viscous riction [ ] A Cross section area o low [m 2 ] di Latent heat [J/kg] q Heat lux, perormance [W] Re Reynolds number [ ] Pr Brandt number [ ] Fr Fraud number [ ] C Coeicient [s/kg 2 /m] ρ Density [kg/m 3 ] δ Thickness [m] d Diameter [m] w Velocity [m/s] i Speciic enthalpy [J/kg] R Gas constant [J/kg K] V Volume [m 3 ] x Vapor quality [-] P Pressure [Pa] Subscripts and superscripts w water rerigerant (Freon) I input o output m middle, normal c critical p parallel v volumetric 1 Introduction The evaporator is the central component o the heat pump. The coeicient o perormance (COP) o the heat pump depends largely on the construction o the evaporator. The construction o the evaporator signiicantly inluences the rerigerant low pattern within the pipes and on the shell side in water. The low pattern o streaming luids greatly aects the value o heat transer coeicient and the value o viscous riction coeicient. 42

3 The quality o the mathematical model o heat pump's evaporator depends largely on the accuracy o the mathematical ormulation o the heat transer coeicient and o the viscous riction coeicient in both streaming luids. The most signiicant eect on the accuracy o the simulation results is the heat transer coeicient o evaporation. In the mathematical model applied governing, equilibrium equations are derived and long since proved. Uncertainty in the model regarding to accuracy represents only the heat transer coeicients and the viscous riction coeicient o rerigerant. In this investigation, we paid special attention to choose the heat transer model o rerigerant evaporation. Among the many models we chose Kandlikar's [1] improved model. Kandlikar's model or the heat transer o rerigerant evaporation gave the best results in the simulation, according to the tendency and according to the intensity. Róbert Sánta [4] in his work has conirmed the justiication o applying the Kandlikar's model. Besides the evaporator construction, the system parameters signiicantly aect the coeicient o perormance (COP) o the heating-cooling system when using heat pumps. We carried out numerical simulation to veriy the accuracy o the mathematical model o the evaporator, and simultaneously investigated the behavior o the evaporator in steady-state operation or various values o water mass low rate as well. In this paper the boundary conditions, which are the thermal expansion valve and the compressor, have not yet been taken into account. It is the task and the goal o the next article. In this paper is a new methodology: implant the evaporative heat transer coeicient as the unction o vapor quality according to Kandlikar [1], apply the viscous riction coeicient o the rerigerant according to Friedel [2] and use the state equations o R134a, according to Martin-Hou [3]. In addition, using the new model the behavior o coaxial evaporator was simulated. 2 Physical Model o the Coaxial Evaporator Structurally, the considered evaporator as a coaxial tube heat exchanger. From thermal aspect, in the evaporator the heat transer is perormed between the rerigerant and the well water. In the observed case, rerigerant Freon R134a lows inside the evaporator tubes, while the cooled mass - the well water - lows inside the shell o evaporator. In the evaporator, the parallel pipes are connected with the bales. The bales are placed perpendicular to the pipe bundle at a distance o about 150 mm in the direction o the tube axis. Water lows between the bales like a sinusoid. 43

4 J. Nyers et al. Mathematical Model o Heat Pumps' Coaxial Evaporator with Distributed Parameters The evaporator works most eiciently when the ull length o parallel pipes is illed with rerigerant which still contains the liquid phase. The quantity o evaporated rerigerant depends on the compressor capacity; the temperature and the mass low rate o well water respectively. The thermo-expansion (TEX) valve doses an adequate quantity o rerigerant into the evaporator. This valve uses the sensors monitors the temperature and the pressure o the outlet rerigerant rom the evaporator. Based on the measured temperature and pressure, the valve carries out optimal dosing o the rerigerant. In case o a rerigerant quantity deicit, the shorter length o the pipes evaporates the liquid phase o rerigerant, and in the remaining part o the evaporator lows only vapor which superheats. The heat transer o the Freon's' vapor phase is multiple times lower than o the liquid phase. The vapor o the rerigerant in the dry section superheats. Over-dosing o the rerigerant means that the liquid phase cannot evaporate and some quantity o the liquid phase leaves the evaporator. This phenomenon reduces the rerigeration eect, or example the perormance o the heat pumps' evaporator. Interpretation: non-evaporated liquid phase evaporates in the compressor. Consequently, hydro hummer can happen in the compressor. In the correct operation mode rom the evaporator dry vapor lows out, superheating up to 4 C. The evaporator is connected to the compressor. From the aspect o the evaporator, the compressor only sucks out the superheated vapor rom the evaporator. From the aspect o compressor, the compressor using mechanical work compresses the vapor which primarily increases the temperature on the adequate level. The compression is unortunately accompanied with intensive increase o the pressure as well. mv tvi mv tvo m ti m to x=0 x=1 x=1 Figure 1 The physical model o the heat pumps' evaporator with the system parameters 44

5 3 Stationary Mathematical Model o the Evaporator 3.1 Conditions and Approximations During the modeling the next conditions and approximations have been applied. Distributed parameters, Operation mode is steady-state, Pressure drop in the rerigerant is taken account, Evaporator walls' thermal resistance is not neglected, Speciic heat o the water is taken as a constant because it slightly depends on the temperature, Speciic heat o the rerigerant vapor is taken as constant because it slightly depends on the temperature, Evaporator's heat gain rom the environment is neglected. 3.2 Governing Equations Balance Equations o the Rerigerant Mass conservation w z = 0 (1) I the operation mode is steady-state wm const. (2) Impulse conservation w w p z Energy conservation + (x)=0 w w w / 2 i z - q(x)= Energy Balance Equation o the Pipe Wall Using the Energy Conservation Law C cv (T w T ) C c c (T T c ) 0 (3) (4) (5) 45

6 J. Nyers et al. Mathematical Model o Heat Pumps' Coaxial Evaporator with Distributed Parameters Energy Balance Equation o the Well Water which Flows through the Evaporator, Based on the Energy Conservation Tw wv z C w T T 0 w c (6) 3.3 Auxiliary Equations State Equations o the Rerigerant R 134a The thermodynamic equations o rerigerant R 134a are taken rom ISO / DIN standards. The equations are applicable at the ollowing conditions: 0.60 < T r < [bar] < p < 25 [bar] T max = 500 [K] The Balance Equation according to Wagner B1 B2 1 A 1 1 T r A 2 1 T r A 3 1 T r lnp r A 1 T B3 A 1 T B4 A 6 (7) T r 4 r 5 r Where: The relative temperature T Tr (8) T C The relative pressure p r p p C (9) The critical temperature o R134a K Tc (10) The critical pressure o R134a pc 40.5 bar The Parameters o the Rerigerant o Saturated Area Speciic enthalpy o the mixture (11) 46

7 i i' x i" i' (12) Speciic internal energy o the mixture u u x u u (13) Speciic volume o the mixture: v = v + x (v v ) (14) State Equation o R134a according to Martin - Hou [3] or the Superheated Area Martin-Hou used Van der Waals equation o real gases and adapted the equation with the correctional coeicients or rerigerants (i.e. or Freon). The equation is applicable or the Freon vapor superheated area. R T p = V b + Where: 5 i=2 The coeicients rom the equation A i + B i T + C i exp( K T Tcrit ) (V b) i (15) b K m kg Speciic Enthalpy o the Freon Vapor i(t r, v) = i o + (p v R T) + D 1 T + D 2 T D 4 lnt + E1 z + E 2 2 z + E 3 3 z + E 4 4 z + e kt R (1 + k T R ) ( G1 z + G 2 2 z 2 + G 4 4 z 4) (16) Pressure Drop in the Single-Phase and the Bi-Phase o Rerigerant (x) For the determination o pressure drop in the rerigerant, Friedel's mathematical model was applied. Friedel's [2] mathematical model is used in the area o single-phase and bi-phase evaporation as well. p 2 dp dz dz (17) 47

8 J. Nyers et al. Mathematical Model o Heat Pumps' Coaxial Evaporator with Distributed Parameters Pressure drop actor o the liquid phases dp dz 2 4 G 2 D Friction actor o the lowing luids was determined according to Blasius Where: For the liquid phase λ Re Re For the vapor phase g 0.25 Re g G D (18) (19) G D Re g (20) g The characteristic riction actor in bi-phase low F H E (21) Fr We Latent Heat as a Function o the Evaporation Temperature di = a o + a 1 t 1 2 o + a 2 t o (22) Where the constants o the rerigerant R 134a are: a o = a 1 = a 2 = Evaporative Heat Transer Coeicient o the Rerigerant For the determination o the evaporative heat transer coeicient o rerigerant inside the pipe Kandlikar's improved mathematical model was used. The same model was used or single-phase and bi-phase rerigerant as well. Improved Kandlikar's model separately estimates the heat transer coeicient o the liquid and the vapor phase. The result is taken as the maximum value o the calculated two. tp max n, c (23) 48

9 3.3.7 Heat Transer Coeicient o the Convective Area Co n 1058 Bo (1 x ) (1 x ) F FL lo Heat Transer Coeicient o the Bubble Area Co c Bo 0.7 (1 x ) (1 x ) lo F FL lo lo Fr Fr lo lo (24) (25) Heat Transer Coeicient in the Single Phase Region according to Petukhov and Popov Interval o the applicability: 0.5 Pr lo 2000 (26) Relo 5 10 (27) lo Relo Prlo d Pr lo 2 (28) 3.4 Equations o the Heat Source Fluid - Well Water The complex low o the water has been taken into account when the heat transer coeicient o the cooled mass, i.e. o well water, which was determined by applying the circular bales. The bales are placed perpendicularly on the evaporators' tubes. The bales orce the water to low in the orm o a sinusoid. The water stream is separated approximately in two sections, parallel with the pipes and perpendicular to the pipes' direction Mathematical Model o the Heat Transer Coeicient o Well Water according to Hausen w w Rew Prw (29) d Reynolds number o cooled mass - well water Re w. G eq d k w k (30) 49

10 J. Nyers et al. Mathematical Model o Heat Pumps' Coaxial Evaporator with Distributed Parameters Mass velocity in the case o an equivalent cross section:. Geq A e 0.5 G pgm (31) A A 0. 5 (32) p m The mass velocity in case o parallel low. m G p w (33) Ap Ap 2 2 D z d k n 4 The mass velocity in case o normal low. Gm A m m w Am Db s d s k (34) (35) (36) 3.5 Conditions and Initial Numerical Values o the Simulation The initial conditions and values o the simulation in stationary operating modes o the evaporator: Rerigerant is R134a Mass low o the Freon m = 0.07 kg / s Temperature o the well water t = 13 C Pressure in the condenser p k = 15 bar Inlet pressure o the evaporator p i = 3 bar Shell tube diameter D = 62 mm Number o the pipes in the evaporator and the pipe diameter 14 x ø8 x 1 mm 3.6 Condition o Linking Two Parts o the Model The mathematical model consists o two parts: the evaporating sector and the superheating sector. Governing equations o both sectors are the same, but the 50

11 physical parameters are dierent. The thermal and the hydraulic parameters are dierent or the rerigerant mixture and the rerigerant vapor respectively. Namely, the evaporating sector the rerigerant is in liquid and vapor aggregate state, while in the superheating sector the vapor is superheated. The condition or the change o physical coeicients in the mathematical model during the simulation is the vapor quality. The value o the vapor quality in the critical position is x = 1. 4 Numerical Method Mathematical model consists o system coupled partial dierential equations with boundary conditions. The boundary conditions are algebraic equations to describe the behavior o thermal expansion valve and the compressor. In this case, the boundary conditions are only numbers, not unctions. For example, the eect o the thermal expansion valve is represented with the value o the rerigerants' input temperature in the evaporator. For solving the mathematical model the numerical recursive procedure Runge- Kutta and iterative Mac Adams were applied. The procedure o Runge-Kutta is used only or getting initial solutions at discrete points, while the procedure Mac Adams converges the approximate initial solutions by iterative manner to the accurate values. Namely, the recursive procedure Runge-Kutta solves only onepoint boundary problems; in our case, the problem is two-points. The system o the dierential equations with two-point boundary conditions is solved by an iterative procedure, one o these is Mac. Adams. The obtained results are numerical and presented visually in graphical orm. Graphics are visible in Figures 2, 3, 4 and 5. 5 Simulations A ew independent variables were investigated to explain the behavior o the heat pump s evaporator. The investigations were carried out using the mathematical model and the numerical method. The independent variables: Rerigerant superheated vapors' outlet temperature. Pipe wall temperature. Well water outlet temperature. The simulation includes the ollowing investigations: 51

12 J. Nyers et al. Mathematical Model o Heat Pumps' Coaxial Evaporator with Distributed Parameters a) Rerigerant convective heat transer coeicient as a unction o the pipe length α = (z) (37) b) Rerigerant temperature in the evaporators' pipes as a unction o the pipe length t o = (z) (38) c) Well water temperature in the evaporator as a unction o the pipe length t wo = (z) (39) d) Evaporators' thermal perormance as a unction o the pipe length q = (z) (40) 6 Simulation Results The obtained results are based on the created mathematical model and numerical methods. These results are presented in the Figures 2, 3, 4 and 5. Figure 2 Rerigerant convective heat transer coeicient as a unction o the pipe length 52

13 Figure 3 Rerigerant temperature in the evaporators' pipes as a unction o the pipe length Figure 4 Well water temperature in the evaporator as a unction o the pipe length 53

14 J. Nyers et al. Mathematical Model o Heat Pumps' Coaxial Evaporator with Distributed Parameters Conclusions Figure 5 Evaporators' thermal perormance as a unction o the pipe length a) Comments on the Mathematical Model In the mathematical model applied governing, equilibrium equations are derived and already proven. Uncertainty in the model regarding to accuracy represents only the heat transer coeicients and the viscous riction coeicient o rerigerant, The stationary mathematical model o the evaporator consists o the system coupled partial dierential equations and the explicit and implicit non-linear algebraic equations, The model is one-dimensional with distributed parameters, or steady-state operation mode, The pressure drop in the coaxial exchanger is taken into account, The evaporator wall's thermal resistance is not neglected, Speciic heat o the water is taken as a constant because it slightly depends on the temperature, Speciic heat o the rerigerant vapor is taken as a constant because it slightly depends on the temperature, The evaporator's heat gain rom the environment is neglected, Solving the mathematical model is possible only numerically, The applied numerical procedures: o To obtain the initial solutions, a recursive one point numerical procedure, Runge-Kutta was used, 54

15 o To obtain the inal solution, the iterative two-point numerical procedure, Mac. Adams was used. The obtained solutions are in numerical orm, The solutions are presented visually in graphic orm. b) Comments on the heat pumps' evaporator The simulations were perormed at: o Constant temperature o the well water inlet, o Constant mass low rate o the well water. c) Conclusions based on the simulation results The accuracy o the mathematical model depends largely on the applied heat transer coeicient o both streaming luids, With increasing the mass low rates in both luids increases the whirling as well this phenomenon causes the increasing o the heat transer coeicient, To a smaller extent, the viscous riction coeicient o both luids also aects the accuracy, Based on the tested ive models, Kandlikar's heat transer coeicient model o the rerigerant proved the most appropriate. At the beginning and at the end o evaporation, the values o the heat transer coeicient correspond to the expectations. Character i.e. the tendency o the coeicient changes along the evaporator is real. It looks like a parabola [4], The evaporation heat transer coeicient o the Freon R134a shows a logical tendency. While decreasing the quantity o the liquid phase, the heat transer coeicient exponentially decreases and achieves the value o heat transer coeicient o superheated vapor (Figure 2), With increasing the well water's mass low rate the consequences: o Decreases the length o evaporation, the result is the length o the evaporator decreases as well, o The rerigeration capacity o the evaporator increases, o The quantity o superheated vapor increases and tends asymptotically towards a constant value. o The temperature o the evaporating rerigerant slightly decreases in the evaporating sector. The temperature decrease is caused by the pressure drop because the viscous riction is present. o By increasing the mass low rate o well water, the water temperature decreases. 55

16 J. Nyers et al. Mathematical Model o Heat Pumps' Coaxial Evaporator with Distributed Parameters New Scientiic Results a) Evaporative heat transer coeicient in the model was implanted as a unction o vapor quality according to Kandlikar [1], b) Viscous riction coeicient o rerigerant was applied according to Friedel [2], c) State equations o rerigerant R134a were used according to Martin-Hou [3], d) In addition, using the new model behavior o heat pumps' coaxial evaporator was simulated (Figures 2, 3, 4 and 5). Reerences [1] Martin J. J., Hou Y. C.: Development o an Equation o State or Gases, AIChEJ, 1955, 1: 142 [2] Satish G. Kandlikar: Heat Transer and Fluid Low in Mini-Channels and Micro-Channels, Mechanical Engineering Department, Rochester Institute o Technology, NY, USA [3] Fridel L.: Improved Friction Pressure Drop Correlations or Horizontal and Vertical Two-Phase Pipe Flow, European Two-Phase Flow Group Meeting, (1979) Ispra, Italy, June, Paper E2 [4] Róbert Sánta: The Analysis o Two-Phase Condensation Heat Transer Models Based on the Comparison o the Boundary Condition Acta Polytechnica Hungarica Vol. 9, No. 6, 2012, pp [5] Yi-Yie Yan, Hsiang-Chao Lio, Tsing-Fa Lin: Condensation Heat Transer and Pressure Drop o Rerigerant R-134a in a plate heat exchanger I. J. o Heat and Mass Transer Vol. 42 (1999) pp [6] Imrich Bartal, Hc László Bánhidi, László Garbai: Analysis o the Static Thermal Comort Equation Energy and Buildings Vol. 49 (2012) pp [7] Garbai L., Méhes Sz.: Energy Analysis o Geothermal Heat Pump with U- tube Installations. IEEE International Symposium CFP 1188N-PRT EXPRES Proceedings, pp Subotica, Serbia [8] Nyers J., Garbai L., Nyers A.: Analysis o Heat Pump's Condenser Perormance by means o Mathematical Model. I. J. Acta Polytechnica Hungarica, Vol. 11, No. 3, pp , 2014 [9] Nagy Károly, Divéki Szabolcs, Odry Péter, Sokola Matija, Vujicic Vladimir:"A Stochastic Approach to Fuzzy Control", I.J. Acta Polytechnica Hungarica, Vol. 9, No. 6, 2012, pp (ISSN: ) [10] László Garbai, Róbert Sánta: Flow Pattern Map or in Tube Evaporation and Condensation, 4 th International Symposium on Exploitation o Renewable Energy Sources, EXPRESS 2012, ISBN: , pp , 9-10 March, Subotica, Serbia 56

17 [11] Kajtár L., Hrustinszky T.: Investigation and Inluence o Indoor Air Quality on Energy Demand o Oice Buildings. WSEAS Transactions on Heat and Mass Transer, Issue 4, Volume 3, October 2008, pp [12] László Kajtár, Miklós Kassai, László Bánhidi: Computerised Simulation o the Energy Consumption o Air Handling Units, 2011, Energy and Buildings, ISSN: (45) pp [13] László Kajtár, Levente Herczeg: Inluence o Carbon-Dioxide Concentration on Human Wellbeing and Intensity o Mental Work. Bp. 2012, Időjárás, Quarterly Journal o the Hungarian Meteorological Service, Vol. 116, No. 2, April-June 2012, pp , ISSN

Optimum of external wall thermal insulation thickness using total cost method

Optimum of external wall thermal insulation thickness using total cost method Jozsef Nyers *, Peter Komuves ** * Obuda University Budapest, Becsi ut 96, 1034 Budapest, Hungary ** V3ME, Subotica-Szabadka,