For the supercritical fluid case, near and above the critical point at an absolute temperature, T, there is no distinguishable phase-change, but we still have a thermally expandable near-perfect gaseous fluid, with Д ~ 1/ T. Thus, to first order only, we have the supercritical flow rate wsc given by

Г 1 )1/3

WSC * W1F fp = W1Fgas

However, the thermal expansion coefficient is, in fact, non-linear with temperature changes near the critical point, as are many other properties, and the Boussinesq approximation is no longer a good approximation. The virial coefficients accommodate this deviation from perfect gas behaviour, but the properties are extremely non-linear near the critical point.

Therefore, for the higher-pressure case we may adopt a numerical analysis, based on iterative integration around the loop of the momentum equation (since mass is also conserved) for varying loop power inputs, using the thermophysical properties of the supercritical fluid as a function of actual thermodynamic state. Thus the general flow variation with major loop parameters (elevations, losses etc.) follows Equation (4) but with a non-linear expansion coefficient.

Taking the necessary boundary condition for parallel channels of constant pressure drop, differentiating the integral form of the mixture momentum equation, we may solve for the critical mass velocity, <G>, when the flow is unstable. The case and result for a heated supercritical flow is quite complex and requires numerical evaluation. For the adiabatic supercritical flow case, the analysis greatly simplifies. After making some algebraic manipulations, the criterion results in an unstable mass velocity given by:

G C m

< G >« C D ’ where Ц is the kinematic viscosity, De the equivalent diameter, and C the constant of

proportionality for the Reynolds number dependency of the friction factor. Thus, the dependency of the friction losses and the viscosity variation are quite important.

To investigate the feasibility of natural convection cooling for the primary circuit of a supercritical water-cooled reactor called, a simple steady-state, natural-circulation program was written, including the full physical variations of the thermophysical properties of supercritical water. With the initial and boundary conditions to the core, the operating pressure and temperature, the circuit resistance coefficients and the elevation difference between the core and the heat exchanger were specified. With an initially assumed flow the analysis iterated around the loop on flow to calculate the steady-state density and enthalpies in the circuit. To understand the parametric trends, many thousands of these calculations were done for different input conditions. The trends are shown in Figure 4 which confirms the expected that for a given inlet temperature the outlet temperature increases with increasing channel power, with decreasing elevation difference between core and heat exchanger, and with increasing circuit loss coefficient.

The effect of the large density and enthalpy changes around the critical temperature can be seen when the inlet temperature is a parameter. Below about 370oC, the outlet temperature/channel power surface is relatively flat (except for high loss coefficient combined with high power), whereas as soon as the inlet temperature exceeds 380oC the outlet temperature rises sharply regardless of the channel power and loss factors. If the fluid enters the core below the critical temperature, it is at relatively high density and low enthalpy, and exits above the critical temperature at low density and high enthalpy. The large density difference gives a large natural-convection driving force: the large enthalpy change allows a high channel power with relatively low flow and pressure drop.

To utilize the maximum design flexibility of elevation and loss coefficients within a maximum outlet temperature limit with a high-powered channel, it is necessary to keep the inlet temperature below the critical temperature and the outlet above the critical temperature. If the inlet temperature is allowed to rise above the critical temperature, the much-reduced density and enthalpy changes result in a very much higher outlet temperature.