FIGURE 2. Control system block diagram (a) for conventional engine-control system in which engine thrust and reactor temperature commands derive from basic vehicle velocity loop (not shown) and (b) advanced system in which reactor power and propellant flow are coupled through reactivity of hydrogen density in reactor core, allowing for much simpler system. Gas temperature (T.) at the nozzle entrance determines specific impulse (Isp) through the relationship I,~(Tc/MW), where MW is the molecular weight of the propellant. A program in which reactor power regulates gas temperature ensures maximum specific impulse without excessive reactor temperature. For dynamic-response reasons and safety consideration subsidiary control loops are closed on pump speed, control rod position and reactor power. Although in this first study we have treated thrust and temperature commands as independent for illustrative purposes, they are, in fact, not independent for the bleed-cycle engine. First, propellant flow rate interacts with reactor power through the reactivity of the hydrogen in the core. This reactivity varies about as the square of flow rate (for constant power) and may be worth of the order of a few dollars ($1 in 8k=0.0065). The second interaction comes about through circumscribed navigation on the pump map. The second control system (fig. 2b) is a much simpler design in which the reactor is designed so that at rated operating conditions the reactivity loss due to the increase in reactor temperature is exactly balanced by the reactivity of hydrogen present in the core. Here only a thrust control loop is neededcalculations and calibrations guarantee in advance that the reactor temperature will not exceed its bounds. (Some variable reactivity would probably be used to achieve cold critical operation of the reactor, but thereafter this reactivity would not be used in the dynamic control of the engine.) System dynamics The dynamic behavior of an automatic control system is described by a set of differential equations relating its principal components and subsystems. The pertinent equations are given in the box for the two systems studied, with definitions for the symbols. We will discuss in a qualitative way the dynamic behavior of the main components of the two systems. Heat exchanger.—Calculations of the heat exchange in the reactor core are complicated by variations in specific heat with temperature, and the dependence of heat transfer coefficients on temperature and gas flow rate. The problem can be simplified by subdividing the cylindrical core into three or more coaxial stages with two walls each. A one-stage model of the core (no axial or radial subdivision), however, illustrates qualitatively the dynamics of the heat exchange process (5, 6). If we linearize the equations for small perturbations from steady state operation, we obtain the transfer functions shown in the box. From Eq. 17, given a small step increase in power with propellant flow held constant, the exit gas temperature varies as (AQ/c,W.)(1—e ̄t/h). If the power is held constant, Eq. 18 shows that a small step decrease in flow rate produces a step increase in gas temperature of amount AWQ0,2/‚xC»W.* followed by a further rise having the above characteristic exponential behavior. The thermal time constant 7 of typical conceptual rocket reactors is of the order of one second. Eq. 19 indicates that TM increases with core mass heat capacity and is inversely proportional to propellant flow rate. Thus a fast reactor, with little moderator mass, may have a time constant of a fraction of a second. Neutron kinetics.—Although the reactivity of the hydrogen in the core is positive, its derivative with respect to reactor temperature is negative. This latter fact makes the neutronics heat-exchanger loop inherently stable and gives rise to the second of its two control systems discussed above. An example will illustrate core response to a change in propellant reactivity: given a step increase in flow (with control-rod reactivity held constant), there is an immediate step decrease in gas temperature, followed by a fairly fast rise in power, which in turn causes a partial recovery (rise) in temperature. The transitions are strongly damped, and both power and temperature are stable at their new equilibrium values. Turbopump feed system.—The liquid hydrogen flow system basically comprises a turbine inlet valve and an axial-flow turbopump. Provided pipe resistance, pipe capacitance, and propellant compressibility are neglected, the system can be represented by a first order differential equation. Pump speed lags behind valve area with a time constant proportional to the moment of inertia of the turbopump rotor divided by the square root of the pump speed (or output flow rate from the pump). The time constant is typically a fraction of a second. The gain of the valve-turbopump combination is proportional to turbine inlet pressure. To provide a control system with overall dynamic response approximately independent of flow rate, a loop is closed on pump speed in which the loop controller has a gain that increases with decreasing speed (11). The response of the turbine inlet valve can be made so fast that it was not included in the study. Thrust__ Design conditions for hypothetical rocket engine Specific impulse (including losses). Reactor power_ Propellant weight flow rate_-_ Nozzle entrance stagnation pressure_. Reflector entrance pressure_. Nozzle exit pressure--- Nozzle throat area. Nozzle expansion ratio___ Gas temperature at reflector entrance_ Gas temperature at core entrance. Neutron mean effective lifetime. Temperature reactivity. Heat-exchange thermal time constant_ Approximate reactor bandwidth (ẞ/l*) Control actuator dynamics, wn, (—— Turbopump speed closed-loop bandwidth_. Controller design (proportional plus integral, k,+k1/8). Thrust control: kr=0.001 sec-1 100,000 lb. 760 sec. 2,260 Mw. 130 lb/sec. 1,100 psia. 4,500° R. 1,220 psia. 4.0 psia. 61 in.' 1.4X10-5 sec. 50 sec1, 0.7. 40 sec-1 kip=0.010 sec-2 Transducers. We assumed thermocouples measure exit gas temperature. Their response lags behind the gas temperature with a time constant that varies directly with the mass of the thermocouple and inversely with flow rate. A typical time constant is a fraction of a second. Lead feedback (with noise filter) to compensate for the lag response provides a fast-acting temperature control loop. Pressure transducers respond rapidly, have little effect on loop dynamics, and were, therefore, not included in the study. Subsidiary control loops.—A power loop based on the logarithm of power has been found to work satisfactorily throughout the power decades in which there is significant heat generation in the reactor (12, 13). Use of the logarithm cancels the customary increase in the gain of the reactor transfer function with power level. The loss in actual power resolution in using the logarithm is not important when the power loop is part of an overall loop closed on reactor temperature. The bandwidth of the power loop is limited by the bandwidth of the control rod drive mechanism. Control study results 235 The engine hypothesized for the control system study possessed the characteristics given in the table. The reactor was assumed to have U2 fuel, graphite moderator, and beryllium reflector. Large-scale transient solutions of the nonlinear equations in the box were obtained from an analog computer, while frequency responses and steady-state solutions were calculated by an IBM-704. Intercomparison of the two calculations led to the elimination of many computational errors. The results presented below are preliminary only in the sense that details of the engine behavior may change somewhat as the characteristics of more individual control components are added to the overall simulation. We believe that the general behavior of the engine is clearly established. An example of the operation of the conventional control system, as simulated by the analog computer, is shown in figure 3a. Here thrust and temperature are commanded to rise linearly from low to full values in an interval of 10 seconds. Reactor temperature rise is made linear with time to minimize internal thermal stress. The turbopump is brought to full-flow operation along a line of constant specific speed, without using the pump bypass valve. The thrust and temperature errors are seen to be quite small fractions of their ultimate values. Both initial errors (undershoots) last longer than the final error transients (overshoots) because of the increase in many of the system time constants at low propellent flow rate. Observe that reactor power is a completely dependent variable-it is entirely determined by the input commands to the system, the closed temperature loop and the interaction of the pertinent variables in the core. Although the terminal power overshoot is considerable it has no serious effect on core temperature. During this startup about 850 pounds of hydrogen were expended to produce an impulse of 5.3X105 lb-sec for the rocket stage. A comparable startup using the simplified control system is shown in figure 3b. With the reactor initially at low power, a command is given for a linear increase in thrust. Propellent reactivity increases the reactor power. Operation stabilizes at full thrust, with final propellent reactivity balancing loss of reactivity from the rise in core temperature. In this mode of operation both reactor power and core temperature are dependent variables. The temperature is seen to rise nearly linearly with time and to arrive at full value 2 to 3 seconds after full thrust is developed. The lack of a closed temperature loop creates smoother demands on reactor power and makes the power response more sluggish than that of figure 3a. Other phases of the study currently in progress concern transient behavior of the engine while in full operation, and the shutdown programs necessary to preserve the reactor for subsequent use. |