A STEADY STATE MODEL FOR THE HEAT PIPE-ENCAPSULATED NUCLEAR HEAT SOURCE

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1 Joint International Workshop: Nuclear Technology and Society Needs for Next Generation Berkeley, California, January 6-8, 2008, Berkeley Faculty Club, UC Berkeley Campus A STEADY STATE MODEL FOR THE HEAT PIPE-ENCAPSULATED NUCLEAR HEAT SOURCE Steven Mullet Department of Nuclear Engineering University of California, Berkeley, Berkeley, CA smullet@berkeley.edu ABSTRACT A major trend in the nuclear industry lately is towards more passive safety systems in power plants. Heat pipes are passive devices that have been used in many industries to remove heat without any electrical input. A design being examined at UC Berkeley is a fast-spectrum heat pipe reactor, called the Heat Pipe-Encapsulated Nuclear Heat Source (HP-ENHS), which utilizes sodium heat pipes to remove heat from a solid molybdenum reactor core fueled by uranium and plutonium nitride. This idea has been examined for space reactor applications before, and with a thermal output of 125MW and the potential to run the reactor with 20 year refueling cycles (from the neutronics data) the HP-ENHS could be suitable for remote sites that are far from the power grid. The focus of this talk will be the ability of the heat pipes to remove heat effectively using a model that predicts the maximum throughput based on capillary limits. Key Words: Heat pipes, advanced reactors, reactor design, passive systems 1. INTRODUCTION Heat pipes are used as heat transfer mechanisms in a variety of ways in many fields. A wide variety of working fluids could be used provided materials are chosen that are compatible and the proper temperature range can be achieved. For high temperatures liquid metals offer excellent heat transfer capabilities. Thus liquid metal heat pipes are well suited for high temperature passive heat removal environments. The particular application proposed is as the primary coolant in a fast-spectrum nuclear reactor. The Heat Pipe-Encapsulated Nuclear Heat Source (HP-ENHS) is a fast reactor that uses heat pipes filled with sodium to remove heat through natural circulation. The heat pipes and fuel rods (of equal diameter) are arranged horizontally in the core which consists of a monolithic rectangular block made of molybdenum TZM. The fuel is uranium nitride and plutonium nitride along with other transuranics. The secondary coolant proposed is a molten salt (2LiF-BeF 2 ) known as FLiBe with very favorable heat transfer and expansion characteristics. This "battery" type nuclear reactor is proposed to operate for approximately 20 years before refueling and is ideal for developing countries, remote sites, and certain industrial applications. Due to its block / rectangular shape at the end of the 20 year cycle the relatively small reactor core could be replaced. A number of heat pipe steady state operating limits have been documented including the sonic limit, the entrainment limit, the boiling limit, and the capillary limit. The capillary heat pipe operating limit is one of the most important of all of the heat pipe power limits. Without the necessary capillary pressure forces the liquid sodium will not be able to return to the evaporator

2 Steven Mullet and the heat pipe will cease to function. With this in mind a model to predict this capillary limit was constructed for sodium heat pipes at high temperatures. Using this model, different wick materials and dimensions can be compared to see which yields the best results Pressure and Temperature Analysis 2. MODEL DESCRIPTION This model simplifies the heat pipe by assigning for main points to examine. In this way a one dimensional analysis is possible by examining the heat transfer and pressure drops from one point to the next. A typical heat pipe has a wick structure for condensed liquid flow surrounding a vapor flow section. In the model there are four points of interest; two points for the wick and vapor passage sections in the evaporator section (denoted with subscripts we and ve ) and two points for the wick and vapor passage sections in the condenser section (denoted with subscripts wc and vc ) as shown in Figure 1. 9 Figure 1. General heat pipe schematic The model does not examine the adiabatic section in any detail and calculates the effective length of the pipe, L eff, by adding half of the condenser and evaporator lengths to the length of the adiabatic section. 1 The flow in the wick is modeled using Darcy s law shown below. (1) Here κ is the permeability of the wick a number which is often determined experimentally. The mass flow,, is derived straight from the power and the enthalpy of vaporization and ρ l represents the density of liquid sodium. The pressure drop for vapor in the center passage comes from the standard equation for flow in tubes. 2/6

3 A Steady State Model for the Heat Pipe-Encapsulated Nuclear Heat Source (2) D v is the vapor flow diameter in the center passage, V v is the velocity of the vapor, and ρ v is the density of sodium vapor. The friction factor, f, is determined by first finding the flow regime through calculating the Reynolds number, Re D, in the vapor passage. Once this is determined the friction factor can be obtained through use of one of the following three equations. 6,7 (0 < Re D < 2000) (3) (2000 < Re D < 20000) (4) (Re D > 20000) (5) An assumption is made that in the condenser the liquid and vapor pressure in the wick and vapor passage are equal such that. (6) In typical heat pipes the pressure drop from vapor to liquid in the condenser is much less than the pressure drop in the evaporator. 1 The final equation comes from the Young-Laplace equation describing the pressure drop in the evaporator. Assuming each pore is a tiny cylinder is can be shown that (7) Here r c is the radius of curvature and σ is the surface tension of the fluid. For simplification it is assumed that the contact angle between the fluid and the pore wall, θ, is zero and the fluid perfectly wets the wall. Equations (1), (2), (6), and (7) can be combined and r c can be solved for as shown. (8) After all of the calculations are complete r c is compared with the known pore radius, r p, to determine if the capillary limit has been reached. If r c < r p the capillary limit has been surpassed since this is not physically possible. The saturation pressure of sodium as a function of saturation temperature can be reasonably approximated using two constants C 0 and C 1. These constants can easily be solved with a thermophysical property table of sodium. With this approximation the saturation temperature is given by In the evaporator it is assumed that bubble nucleation occurs within a superheated liquid. This assumption works for evaporator regions closer to the center of the reactor, but since the (9) Joint International Workshop: Nuclear Technology and Society Needs for Next Generation Berkeley, California, January 6-8, 2008, Berkeley Faculty Club, UC Berkeley Campus 3/6

4 Steven Mullet evaporator is fairly long it is assumed to be valid. Now combining (9) with the homogenous nucleation equation 3 and solving for yields the temperature of the vapor in the evaporator. (10) 2.2. Heat Transfer Analysis The heat transfer part of the model begins by varying the temperature of the liquid salt secondary coolant. Previous studies performed at UC Berkeley 8 found that with an assumed power throughput of kw per heat pipe in a bundle of 3704 (on one side of the reactor), the range of temperatures in the FLiBe will fall between 832 K and 945 K. The average bulk coolant temperature in the bundle was calculated as 888 K with an average heat transfer coefficient from wall to coolant of 4700 W/m 2 K. All physical data are interpolated when necessary; including properties of liquid and vapor sodium 4 and the thermal conductivities for the various solid materials compared. 5 The thermal conductivity is determined using a highly idealized model that does not account for the specific geometry of the wick. The conductivities are calculated assuming parallel and series heat flow in the sold and liquid portions of the wick separately. (11) These two effective conductivities, k eff, are then averaged to obtain an estimate of the effective thermal conductivity. Here φ is the porosity of the wick and is assumed to be The conductivities k s and k l represent the conductivities of the solid and liquid respectively. In practice a thermal insulating layer could be employed outside the heat pipe in the condenser to ensure that the heat pipe coolant stays at the constant design temperature. 2 A simple one dimensional analysis is used to evaluate the conductive and convective heat transfer to and from the heat pipe in the condenser to start the calculation. The coolant temperature is assumed and thermal resistances are calculated based on the heat transfer coefficient to the coolant, the conductivity through the molybdenum wall, and the effective wick conductivity. Basic heat transfer calculations 6 for the thermal resistance of a cylinder are employed to calculate the temperature of the sodium in vapor passage of the condenser. The physical properties are recalculated at this temperature and the process of calculating the various pressure drops and temperatures can proceed. 3. PRELIMINARY RESULTS One of the most difficult parts of this model is obtaining data for certain wick characteristics including the pore radius and the wick permeability, For this reason actual wicks are not used in (12) 4/6

5 A Steady State Model for the Heat Pipe-Encapsulated Nuclear Heat Source this preliminary analysis, instead wicks are assumed to be able to be fabricated from several compatible materials with a porosity, permeability, and pore radius that is typical for wicks in this field. The plots are therefore somewhat hypothetical and more data on specific wick designs will certainly improve this comparison. For these reasons Figure 2 shows several cases with varying values for permeability with molybdenum as the wick material. This demonstrates just how sensitive the capillary limit is to changes in permeability. A porosity of 0.65 is very common for heat pipe wicks so it is used here as well. Previous work has discovered that the pore radius can be manufactured with lasers to as low as 2.5 μm. Since this work is for an advanced reactor design many years away from the manufacturing stage it is assumed that such a pore radius can be obtained. Physical characteristics of the heat pipe were chosen from previous studies at UC Berkeley 8 : the outer heat pipe diameter is 1.56 cm, the inner heat pipe diameter is 1.36 cm, the wick thickness is 0.5 mm, the evaporator length is 75 cm, the adiabatic length is 50 cm and the condenser length is 48.8 cm. The temperature in the secondary coolant, T FLiBe, varies from 800 K to 1000 K and the power limit is solved at each of these temperatures to create Figure 2. Figure 2. Capillary power limits with varying permeability 4. CONCLUSIONS The shape of the capillary limit curve is a bit suspect in comparison to some of the other capillary limit curves in the literature. However the power limit predicted for the heat pipe near 1150K seems reasonable. Since this is the region of interest in the HP-ENHS reactor this power limit will be acceptable until further more detailed studies can be done. Further studies will analyze the decay heat transfer from the heat pipes to the secondary fluid and to the Reactor Vessel Air Cooling System (RVACS) through the heat pipe block and the vessel walls. This heat pipe steady state model may be incorporated into these studies. Joint International Workshop: Nuclear Technology and Society Needs for Next Generation Berkeley, California, January 6-8, 2008, Berkeley Faculty Club, UC Berkeley Campus 5/6

6 Steven Mullet REFERENCES 1. G.P. Peterson, An Introduction to Heat Pipes, John Wiley & Sons, Inc., New York (1994). 2. C.C. Silverstein, Design and Technology of Heat Pipes for Cooling and Heat Exchange, Hemisphere Publishing Corp., Washington USA (1992). 3. V.P. Carey, Liquid-Vapor Phase-Change Phenomena, 2 nd Ed., Taylor & Francis Group, New York (2007). 4. International Nuclear Safety Center, 5. A. Faghri, Heat Pipe Science and Technology, Taylor & Francis, New York (1995). 6. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New York (2006). 7. W.M. Rosenow and H. Choi, Heat, Mass, and Momentum Transfer, Prentice-Hall, Engelwood Cliffs, New Jersey USA (1961). 8. E. Greenspan, Solid-Core Heat-Pipe Nuclear Battery Type Reactor, NERI Grant Number DE-FC07-05ID Los Alamos National Laboratory, 6/6