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Heated surface

Insulated surface

FIGURE 8-21. Typical temperature distributions through a wall of an uncooled metal thrust chamber as a function of heating time.

equilibrium condition of the wall before the rocket operates; the various curves show the temperature profile across the wall at successive time intervals after initiation of combustion. The line at T = 357°C shows an equilibrium temperature of the wall a finite time after cutoff.

The heat transferred across the hot surface of the wall (and distributed within the wall by conduction) must be less than the heat-absorbing capacity of the wall material below the critical temperature. If heat transfer to the outside atmosphere and axially within the metal wall is neglected, this can be expressed in a very simplified form:

where Q is the heat per second transferred across area A. Eq. 8-17 shows that QjA depends on the hot gas temperature, the wall temperature, and the gas film coefficient. The heat conductivity k depends on the material and its temperature; AT denotes the average wall temperature increment; dT/dL the temperature gradient of the heat flow near the hot wall surface in degrees per unit thickness; m the mass of a unit area of wall; c the average specific heat of the wall material; and At at the time increment. The chamber and nozzle walls can be divided into cylindrical or conical segments, and each wall segment in turn is divided into an arbitrary number of axisymmetric concentric layers, each of a finite thickness. At any given time the heat conducted from any one layer of the wall exceeds the heat conducted into the next outer layer by the amount of heat absorbed in raising the temperature of the particular layer. This iterative approach lends itself readily to two- or three-dimensional computer analysis, resulting in data similar to Fig. 8-21. It is usually sufficient to determine the heat transfer at the critical locations, such as in the nozzle throat region.

A more complex three-dimensional analysis can also be undertaken; here the wall geometry is often more complex than merely cylindrical, heat is conducted also in directions other than normal to the axis, temperature variable properties are used, boundary layer characteristics vary with time and location, and there may be more than one material layer in the wall.

A number of mathematical simulations of transient heat transfer in ablative materials have been derived, many with limited success. This approach should include simulation for the pyrolysis, chemical decomposition, char depth, and out-gassing effects on film coefficient, and it requires good material property data. Most simulations require some experimental data.

Steady-State Transfer to Liquids in Cooling Jacket

The term regenerative cooling is used for rockets where one of the propellants is circulated through cooling passages around the thrust chamber prior to the injection and burning of this propellant in the chamber. It is really forced convection heat transfer. The term regenerative is perhaps not altogether appropriate here, and it bears little relation to the meaning given to it in steam turbine practice. It is intended to convey the fact that the heat absorbed by the coolant propellant is not wasted but augments its initial temperature and raises its energy level before it passes through the injector. This increase in the internal energy of the liquid propellant can be calculated as a correction to the enthalpy of the propellant (see Chapter 5). However, the overall effect on rocket performance is usually very slight. With some propellants the specific impulse can be 1% larger if the propellants are preheated through a temperature differential of 100 to 200°C. In hydrogen-cooled thrust chambers and in small combustion chambers, where the wall-surface-to-chamber volume ratio is relatively large, the temperature rise in the regenerative coolant will be high, and the resulting increase in specific impulse is sometimes more than 1%.

The behavior of the liquid film is critical for controlling the wall temperatures in forced convection cooling of rocket devices at high heat fluxes (see Table 8-4 and Refs. 8-14 and 8-15). At least four different types of film appear to exit, as can be interpreted from Fig. 8-22. Here the heat transfer rate per unit of wall surface is shown as a function of the difference between the wall temperature on the liquid side Twl and the bulk temperature of the liquid Th

1. The normal forced convection region at low heat flux appears to have a liquid boundary layer of predictable characteristics. It is indicated by region A—B in Fig. 8-22. Here the wall temperature is usually below the boiling point of the liquid at the cooling jacket pressure. In steady-state heat transfer analysis the liquid film coefficient can be approximated by the usual equation (see Refs. 8-10 and 8-12):

FIGURE 8-22. Regimes in transferring heat from a hot wall to a flowing liquid.

FIGURE 8-22. Regimes in transferring heat from a hot wall to a flowing liquid.

where m is the fluid mass flow rate, c its average specific heat, A the cross-sectional flow area, D the equivalent diameter of the coolant passage cross section,* v the fluid velocity, p the coolant density, fi its absolute viscosity, and k its conductivity. Many liquid-cooled rocket devices operate in this regime of heat transfer. Values of the physical properties of several propellants are given in Tables 8-5 and 7-1. In Table 8-5 it can be seen that hydrazine is a good heat absorber, but kerosene is poor.

2. When the wall temperature Twl exceeds the boiling point of the liquid by perhaps 10 to 50 K, small vapor bubbles form at the wall surface. These small, nuclei-like bubbles cause local turbulence, break away from the wall, and collapse in the cooler liquid. This phenomenon is known as nucleate boiling. The turbulence induced by the bubbles changes the character of the liquid film and, augmented by the vaporization of some of the propellant, the heat transfer rate is increased without a proportional increase in the temperature drop across the film, as can be seen by the steep slope B-C of the curve in Figure 8-22. If the pressure of the fluid is raised, then the boiling point is also raised and the nucleate

TABLE 8-5. Heat Transfer Characteristics of Several Liquid Propellants

Boiling

Characteristics Nucleate Boiling Characteristics

Boiling Critical Critical

TABLE 8-5. Heat Transfer Characteristics of Several Liquid Propellants

Boiling

Characteristics Nucleate Boiling Characteristics

Boiling Critical Critical

Liquid Coolant

Pressure (MPa)

Temp. (K)

Temp. (K)

Pressure (MPa)

Temp. (K)

Pressure (MPa)

(MW/m2)

Hydrazine

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