Reactor Temperature Coefficients

The temperature dependence of the material density and reactor size are easily defined in the case of bare homogeneous reactors. A change in the temperature produces, of course, a change in the various nuclear densities which are involved in the macroscopic cross sections. Moreover, the thermal expansions which accompany increases in temperature influence directly the physical size of the system and therefore affect the neutron leakage. 

The temperature coefficient is computed by taking the temperature derivative of this relation thus, 

This equation gives the fractional change in k due to a unit change in temperature T. The total change due to a change in temperature dT is clearly 

It follows that in terms of the reactivity the temperature coefficient takes the form 

Note that, in the event that the reactor was initially at steady state 

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One of the contributing factors to a thermal.reactor temperature coefficient is that due to different temperature dependence of fission and nonfission cross sections. It is the purpose of this work to examine this coefficient for the BER,.both initially and after equilibrium plutonium-239 has been attained. While the numerical results are valid only for BER, the qualitative aspects of the problem are applicable to similar reactors. 

The operation of a reactor in the practical sense requires a small temperature coefficient in order that criticality can be easily maintained. Usually in addition to its small magnitude, the overall reactor temperature coefficient is designed to be negative to increase stability and to enhance the safety of the reactor. 

In this experiment the overall reactor temperature coefficient is measured. This is done by heating the water moderator with electrical heaters and monitoring the water temperature. The reactivity due to the temperature change is determined from the measured movements of the calibrated control rod that are required to maintain criticality. 

No inherent instability in plant response to planned and unplanned transients has been revealed, with either a single Brecon or multiple units, assuming the overall reactor temperature coefficient is negative and the Braytons are operating with active speed control. 

Most of the Section 11 casualty design events are included in Table 16-2. Section 11 defines a casualty design event as a transient that the SNPP is designed to survive without permanent loss of capability. The transient results described in this section suggest that this requirement can be met. This conclusion is reached assuming that the SNPP will have a negative reactor temperature coefficient (as expected), a Brayton shaft speed control system and typical reactor protection features. Typical reactor protection may include a means to rapidly reduce reactivity (shim or scram). It may also be advantageous to use a reactivity control system based on Brayton coolant temperature. 

Nonferrous alloys account for only about 2 wt % of the total chromium used ia the United States. Nonetheless, some of these appHcations are unique and constitute a vital role for chromium. Eor example, ia high temperature materials, chromium ia amounts of 15—30 wt % confers corrosion and oxidation resistance on the nickel-base and cobalt-base superaHoys used ia jet engines the familiar electrical resistance heating elements are made of Ni-Cr alloy and a variety of Ee-Ni and Ni-based alloys used ia a diverse array of appHcations, especially for nuclear reactors, depend on chromium for oxidation and corrosion resistance. Evaporated, amorphous, thin-film resistors based on Ni-Cr with A1 additions have the advantageous property of a near-2ero temperature coefficient of resistance (58). 

Figure 1. Typical reactor temperature profile for continuous addition polymerization a plug-flow tubular reactor. Kinetic parameters for the initiator 1 = 10 ppm Ea = 32.921 kcal/mol In = 26.492 In sec f = 0.5. Reactor parameter [(4hT r)/ (DpCp)] = 5148.2. [(Cp) = heat capacity of the reaction mixture (p) = density of the reaction mixture (h) = overall heat-transfer coefficient (Tf) = reactor jacket temperature (r) = reactor residence time (D) = reactor diameter].

Equation (8) provides a general relationship between the reactor temperature profile and the operating parameters. In relating the system heat transfer to the conversion-molecular weights relationship for a reactor of fixed size, the heat transfer coefficient emerges as the correlating parameter. 

This research used mechanically agitated tank reactor system shown in Fig. 1. The reactor, 102 mm in diameter and 165 mm in height, was made of transparant pyrex glass and was equipped with four baffles, 120 mm in length and 8 mm in width, and six blades disc turbine impeller 45 mm in diameter and 12 mm in width. The impeller was rotated by electric motor with digital impeller rotation speed indicator. Waterbath thermostatic, equipped with temperature controller was used to stabilize reactor temperature. Gas-liquid mass transfer coefficient kia was determined using dynamic oxygenation method as has been used by Suprapto et al. [11]. 

The above equation then represents the balanced conditions for steady-state reactor operation. The rate of heat loss, Hl, and the rate of heat gain, Hq, terms may be calculated as functions of the reactor temperature. The rate of heat loss, Hl, plots as a linear function of temperature and the rate of heat gain, Hq, owing to the exponential dependence of the rate coefficient on temperature, plots as a sigmoidal curve, as shown in Fig. 3.14. The points of intersection of the rate of heat lost and the rate of heat gain curves thus represent potential steady-state operating conditions that satisfy the above steady-state heat balance criterion. 

Heat transfer capacity coefficient Exothermic heat of reaction Feed temperature Reactor temperature Concentration Collision frequency Activation energy Gas constant... 

Note that now Tj is a variable that is a function of position Zc in the cooling coif while T, the reactor temperature in the CSTR reactor, is a constant. We can solve this differential equation separately to obtain an average coolant temperature to insert in the reactor energy-balance equation. However, the heat load on the cooling coil can be comphcated to calculate because the heat transfer coefficient may not be constant. 

Significant advances have also been made in reactor safety. Earlier reactors rely on a series of active measures, such as water pumps, that come into play to keep the reactor core cool in the event of an accident. A major drawback is that these safety devices are subject to failure, thereby requiring backups and, in some cases, backups to the backups The Generation IV reactor designs provide for what is called passive stability, in which natural processes, such as evaporation, are used to keep the reactor core cool. Furthermore, the core has a negative temperature coefficient, which means the reactor shuts itself down as its temperature rises owing to a number of physical effects, such as any swelling of the control rods. 

In the new designs, if coolant were lost, the nuclear chain reaction would be terminated by the reactor s negative temperature coefficient after a modest temperature rise. Core diameter of the modular units would be limited so that decay heat could be conducted and radiated to the environment without overheating the fuel to the point where fission products might escape. Thus, inherent safety would be realized without operator or mechanical device intervention. 

An often-used method for the limitation of the heat release rate is an interlock of the feed with the temperature of the reaction mass. This method consists of halting the feed when the temperature reaches a predefined limit. This feed control strategy keeps the reactor temperature under control even in the case of poor dynamic behavior of the reactor temperature control system, should the heat exchange coefficient be lowered (e.g. fouling crusts) or feed rate too high. 

Before we leave this example, let us take a look at the issue of heat transfer. In setting up the simulation, we have specified the reactor temperature (430 K) and volume (100 m3) but have said nothing about how the heat of reaction is removed. The simulation calculates a heat removal rate of 12.46 x 106 W. If the aspect ratio of the vessel is 2, a 100-m3 vessel is 4 m in diameter and 8 m in length, giving a jacket heat transfer area of 100.5 m2. If we select a reasonable 30 K differential temperature between the reactor and the coolant in the jacket, the jacket temperature would be 400 K. Selecting a typical overall heat transfer coefficient of 851 W K-1 m-2 gives a required heat transfer area of 488 m2, which is almost 5 times the available jacket area. Aspen Plus does not consider the issue of area. It simply calculates the required heat transfer rate. 

Figure 3.1 gives a Matlab program that sizes the reactor given the conversion, reactor temperature, feed conditions, coolant properties, and kinetic parameters. Then the coefficients of the linear model are evaluated, and the poles and zeros of the openloop transfer function are calculated. If any of the poles have positive real parts, the system is openloop-unstable. 

The final disturbance tested is a 10% decrease in the overall heat transfer coefficient U. Figure 3.19 shows that the 350 K case with 85% conversion goes unstable. Figure 3.20 shows that the 330 K cases are both stable, but the peak deviation in reactor temperature is over 3 times greater for the 85% conversion process than for the 95% conversion process. 

Note that the b2i coefficient is negative when the feed temperature T0 is less than the reactor temperature TR. This produces a positive root of the numerator polynomial given in Eq. (3.47), so the openloop transfer function has a positive zero. 

Increasing recycle flow reduces the inlet, peak, and exit temperatures of the reactor. Pressure builds until the higher partial pressures of the reactants compensate for the lower specific reaction rate because of the lower temperatures. The higher velocities in the reactor tubes also increase the heat transfer coefficient, which means that the heat transfer rate does not decrease directly with the decrease in reactor temperatures. Remember, steam pressure (and temperature) is held constant in the openloop run. The net result of the various effects is that, with the fresh feed flowrates fixed, the reactor comes to a new steady-state condition, which has lower reactor temperatures but higher pressure. The net reaction rate and the heat transfer in the reactor remain the same. The... 

Figure 7.21 Evolution of the reactor temperature in the case of switching between operating points II and III, in the presence of a 10-K unmeasured increase in the coolant inlet temperature and an unmodeled 20% drop in the heat transfer coefficient U. Both disturbances occur at t = 60 min.

Figure 7.27 Evolution of the reactor temperature and temperature setpoint for a 10% rise in the production rate at operating point I, under plant-model parameter mismatch. The heat transfer coefficient U in the controller model is overestimated by 10% compared with its value in the plant.

For predesign calculations we consider a gas velocity of 0.5 m/s [4], Mean physical properties for the above reaction mixture are density 11.5 kg/m3, viscosity 1.5 x 10 5 N s/m2, thermal conductivity 2.9 x 1(T2 W/m K. The calculation of the heat-transfer coefficient follows the relations given in Chapter 5. Applying the relation (5.9) leads to Rep = 2090 and Nu = 412, from which the partial heat-transfer coefficient on the gas side is a = 3 50 W/m K. Taking into account other thermal resistances we adopt for the overall heat-transfer coefficient the value 250 W/m2 K. For the cooling agent we consider a constant temperature of 145 °C, which is 5 °C lower than the inlet reactor temperature. This value is a trade-off between the temperature profile that avoids the hot spot and the productivity. 

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