Figure 1. Physical model of the heat pump with liquid-suction exchanger

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1 Steady-state Mathematical Model of the Liquid- Suction Heat Exchanger of Heat Pump with R134a Jozsef Nyers *, Daniel Stuparic **, Dorottya Boros ** * University Obuda, Budapest Hungary ** BME, Budapest Hungary nyers@uni-obuda.hu Absrtract This paper presents a stationary mathematical model of the liquid-suction heat exchanger of the heat pump with the concentrated state variables. Model contains the basic equations based on the heat balance and auxiliary equations for heat transfer of the superheated vapor and condensate as well as the state equation for the refrigerant. The model is solved analytically and numerically the obtained numerical results are presented graphically. The main objective was to analyze the obtained heat flow between the condensate and the superheated vapor as a function of the refrigerant mass flow rate. An additional objective was to determine the outlet temperature of the condensate and superheated vapor. INTRODUCTION For heating systems with the heat pump the energy efficiency plays a very important role. Energy efficiency of the heating system increases with decreasing the temperature of water in the heating loop, it can achieve with increasing the surface of heaters. Large heating surface is possible to form in the panels for example primarily in the floor then in the wall and in the ceiling. However often in the application are radiators and fancoils with the smaller heating surface. Due to the reduced heating surface, heating water temperature increases and as consequence reduces the energy efficiency, i.e. COP. Some compensation of the COP deterioration can be achieved by applying liquid-suction exchanger. The liquid-suction exchanger is more effective when the temperature of the heating water increases. The heat pump with CO2 refrigerant without liquid-suction exchanger cannot function. Besides the benefits liquidsuction exchanger has drawbacks: 1. Vapor temperature after compression can achieve a very high value 2. High temperature of vapor has a practical limit. 3. The temperature limit of the superheated vapor limits the sub cooling of the condensate. 4. Use of the liquid-suction heat exchanger may cause oscillations in the flow of the refrigerant. Regulator, TEV can become unstable. The main goal of this research is to determine and examine the heat flux between the condensate and superheated vapor depending on the mass flow rate of the refrigerant. Additional aim is to determine the outlet temperature of the condensate and the superheated vapor. PHYSICAL MODEL The considered physical system consists of evaporator, condenser, compressor, controller TEV, and liquidsuction heat exchanger. Practically, the traditional refrigerating machine is expanded with the liquid-suction exchanger. In liquid-suction heat exchanger is exchanged the heat between the condensate and superheated vapor. The aim is to the optimum level sub cool the condensate. Figure 1. Physical model of the heat pump with liquid-suction exchanger Figure 2. Physical model of the liquid-suction exchanger with variables PROCESS IN THE STATE DIAGRAM In the state diagram, logp-i, of the refrigerant R134a exactly can see quantity and location of the heat transfer between the sub-cooled condensate and superheated refrigerant vapor. The aim of the condensate sub-cooling is the COP improvement of the heat pump. The lack of the process is beside increases the COP after compression increases the superheated vapor temperature, as well. Between the state of 4'-4 takes place sub-cooling of condensate, while between 1 "-2 superheating of the refrigerant vapor. 15

2 Figure 3. Liquid-suction process in logp-i diagram MATHEMATICAL MODEL Introduction A mathematical model was created for the steady-state regime of the heat pump. The equations in the model are algebraic with lumped parameters and variables. The governing equations were derived based on the heat balance for the condensate, the superheated vapor and for the heat flux through the wall of the plate exchanger. Auxiliary equations are: coefficient of convective and conductive heat transfer, specific heat and dynamic viscosity of the condensate and vapor. The accuracy of the obtained results largely depends on the accuracy of the model for the convective heat transfer coefficient. Governing equations By governing equations is determined the heat flux of the condensate, through the exchanger wall on the superheated vapor. At the steady-state operation mode the heat flux is constant through the each layer. = = = (1) Heat flux of the refrigerant superheated vapor = (2) Heat flux through the exchanger's wall = (3) Heat flux of the refrigerant condensate = (4) In case of the steady-state operation mode the mass flow rate is constant that means the mass flow rate of the condensate and the superheated vapor is equal. = = (5) Auxiliary equations The logarithmic temperature difference in the case of the parallel flow, = / (6) The logarithmic temperature difference in the case of the counter flow, = / (7) The overall heat transfer coefficient The wall of the plate heat exchanger is usually of the steel and very thin only mm for this reason the thermal resistance can be neglected. + (8) Heat transfer coefficient of the condensate and the vapor for the plate heat exchangers from Holger Martin [7] is., = $,, 2/3 ( % ) *+ &',, -.2/ 01, &' Where: Reynolds number of the one channel is. (9) *+ &',, = % &' &' 4, (10) Equivalent mass flow rate in the one channel &' = / (11) Hydraulic diameter of the channel suggested by Shah and Wanniarachchi, d ekv, is defined as % &' = 2 6 (12) Mean channel spacing, b, is defined as the pitch, p, minus thickness of the plate, t. 6 = 7 β - angle of the chevron f :;< - Fanning friction factor Prandtl number 4,, 01, = $, (13) Surface of the liquid-suction plate heat exchanger = = > (14) Channel flow area is determined as the mean channel spacing, b, multiplied by width, B. 5 = 6 > (15) Physical parameters of the refrigerant R 134a [2] The temperatures conversion from [ ] to = = D (16) Coefficient of the saturated-liquid thermal conductivity $ = 2, F K/, D (17) Coefficient of the saturated-vapor thermal conductivity $ = F, K/, D (18) Specific heat of the saturated-vapor (R134a from -10C to +76C) 16

3 , = F2/,.H,O332N2PJ2 Q/RD (19) Specific heat of the saturated-liquid valid for (R134a from -10C to +76C), = F2/S.O,3NPHNPM32 Q/RD (20) Coefficient of the dynamic viscosity of saturated-liquid, valid for (R134a from 260 K to 350 K) 4 = 10 FM FH 0T - (21) Coefficient of the dynamic viscosity of saturated-vapor, valid for (R134a from 260 K to 350 K) 4 = 10 FM , FM 0T - (22) ANALITICAL PROCEDURE Analytical solution of the governing equations is possible only with approximation of the logarithmic temperature difference to the arithmetic mean value. Deviation due to approximation is only 2-3%, because during the heat transfer the temperature variation is slightly nonlinear. The arithmetic mean value of the parallel and counter flow is: Parallel flow. UVW + U = 2 = 1/2 + = Counter flow. = 1/2 + UVW + U = 2 = 1/2 + = (23) = 1/2 + (24) It can be seen, the arithmetic mean value of the parallel and the counter flow exactly the same. The known i.e. the input independent variables are the input temperature of the condensate and superheated refrigerant vapor. The unknown i.e. the output temperature can be expressed as a function of the known input temperature and replace them in the relation of the arithmetic temperature difference. Using the equation (1) can be obtained outlet temperature of the superheated vapor. = +, (25) Using the equation (3) can be obtained outlet temperature of the condensate. 17 =, (26) Substituting the terms of the outlet temperatures (25), (26) and the arithmetic temperature difference (24) in the equation (2), the heat flux through the exchanger wall is obtained as a function of the input-known temperatures. = 1/2 + 2, + X Y Z[ U X [ Y Z\ U \ (27) Because of steady-state regime the mass flow rate is constant and shares by channels. Channels number of the condensate and vapor side is same, fifty-fifty of the total number of channels n/2. In an ideal case, by the same ratio is divided the mass flow rate. = = (28) The analytical relation of the heat flux in the final form is. 2 ] F2 ^ (29) NUMERICAL PROCEDURE Introduction Mathematical model of the liquid-suction heat exchanger contains coupled, nonlinear, explicit and implicit algebraic equations. Analytical approximate solution is possible only by negligence. Without negligence, to obtain the solution is possible only numerically applying one of the numerical methods from mathematical software package "Matlab". Variables analyzes Independent variables in the considered system are the input temperature of the condensate, (t ci ), and the saturated vapor, (t vi ), heat exchanger surface area, (F) or number of plat, and the mass flow rate of the refrigerant,. Dependent variables are condensate output temperature, (t co ), and the superheated vapor temperature, (t vo ). CASE STUDY The behavior simulation of the liquid-suction heat exchanger was performed by applying the presented mathematical model with the following data. Input data i.e. independent variables: Refrigerant is R134a Input temperature of the condensate t ci = 50; 60 [ o C] Input temperature of the superheated vapor t vi = 0 o C mass flow rate of the refrigerant m = 0.01; 0.02; 0.03 [kg/s] Physical parameters of the plate heat exchanger: Length, L = 470 mm Width, B = 115 mm Number of plate, n = 2; 4; 6; 8; 10. Plate thickness t = 0.4 mm

4 Mean channel spacing b = 2 mm Physical parameters of the R134a: Coefficient of the saturated-liquid thermal conductivity (17) Coefficient of the saturated-vapor thermal conductivity (18) Specific heat of the saturated-vapor (19) Specific heat of the saturated-liquid valid for (20) Coefficient of the dynamic viscosity of saturatedliquid, valid for (R134a from 260 K to 350 K) (21) Coefficient of the dynamic viscosity of saturatedvapor, valid for (R134a from 260 K to 350 K) (22). Remarks: The specific heat of the sub-cooled condensate or of the superheated vapor, C p, is calculated using the correlations of the saturated condensate or of the superheated vapor. Due to approximation the maximum deviation is less as 1%. Output data i.e. dependent variables: Condensate outlet temperature, t co, Outlet temperature saturated vapor, t vo, Heat flux, q. Output data may be as a function of the exchanger's surface, refrigerant mass flow rate, inlet temperature of the condensate and the superheated vapor. VIII RESULTS Figure 4. Heat transfer variation as a function of the refrigerant mass flow rate. q(m)-arithmetic model, q e(m)-parallel flow, q p(m)-counter flow by use the logarithmic model. CONCLUSION In the paper is presented a stationary lumped parameters mathematical model for describing the behavior of heat liquid-suction exchangers in the heat pump. The model is solved analytically and numerically. Numerical solution has been done, for parallel and counter flow of condensate and superheated vapor. Analytical solution is obtained by approximating the logarithmic with the arithmetic temperature difference. Correlation is not different for parallel and the counter flow because in this case the arithmetic mean temperature difference is same for both flows. Numerically obtained simulation results show that: The results of analytical model are closer to the results obtained using the counter- than parallel flow numerical model. The choice of the surface size of the liquid-suction heat exchangers, possible to set the desired output temperature of the superheated vapor. Figure 4. As an alternative solution of the temperature regulation is applying the by-pass for regulating the mass flow of the condensate or the superheated vapor. The outlet temperature of the superheated vapor is limited by the outlet temperature of the superheated vapor after compression. REFERENCES A. Laesecke, R.A. Perkins, and C.A. Nieto de Castrob:" Thermal, conductivity of R134a" Fluid Phase Equilibrium, 80 (1992) pp , Elsmier Science publishers B.V., Amsterdam. Du Point's thermodynamic properties of the R134a, Scalabrina and P. Marchi: "A Reference Multi parameter Viscosity Equation for 134a with an Optimized Functional Form". J. Phys. Chem. Ref. 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