## Mda

Hot gas

Coolant fluid films Inner wall Gas film Coolant fluid

Coolant fluid films Inner wall Gas film Coolant fluid Z // Ambient air rn VA— temperature ust chamber (294 K)

Radial distance from center line of thrust chamber

Z // Ambient air rn VA— temperature ust chamber (294 K)

FIGURE 8-20. Temperature gradients in cooled rocket thrust chamber. The listed temperatures are typical.

can be used in these equations. These simple relations assume that the heat flow is radial. The simple quasi-one-dimensional theory also often assumes that the thermal conductivity and the film coefficients are at average values and not functions of temperature or pressure. A two- or three-dimensional finite element model would also need to be used to analyze the heat transfer in the axial directions, which usually occurs in the nozzle throat wall regions; some of the heat from the hot nozzle insert is transferred to wall regions upstream and downstream of the insert.

Because the film coefficients, the gas and liquid coolant temperatures, the wall thickness, and the surface areas usually vary with the axial distance within a combustion chamber (assuming axial heat transfer symmetry), the total heat transfer per unit time Q can be found by integrating the local heat transfer over the entire internal surface area of the chamber and the nozzle:

Because both q and D are complicated functions of L, the equation usually has to be solved by dividing the rocket chamber into finite lengths. Assuming that q is given by Eqs. 8-15 to 8-19 and remains constant over the length of each element gives an approximate solution.

The important quantities for controlling the heat transfer across a rocket chamber wall are the fluid film boundaries established by the combustion products on one side of the wall and the coolant flow on the other. The gas film coefficient largely determines the numerical value of the heat transfer rate, and the liquid film largely determines the value of the wall temperatures. The

determination of the film coefficients in Eqs. 8-17 and 8-19 is difficult because of the complex geometries, the nonuniform velocity profile, the surface roughness, the boundary layer behavior, and the combustion oscillations.

Conventional heat transfer theory is usually given in terms of several dimen-sionless parameters (Ref. 8-10):

where hg is the film coefficient, D the diameter of the chamber of the nozzle, v the calculated average local gas velocity, k the conductivity of the gas, ¡i the absolute gas viscosity, cp the specific heat of the gas at constant pressure, and p the gas density.

In Eq. 8-21 the quantity hgD/ic is known as the Nusselt number, the quantity Dvp/fi as the Reynolds number, and the quantity cp/x/k as the Prandtl number Pr. The gas film coefficient hg can be determined from Eq. 8-21:

where pv is the local mass velocity, and the constant 0.026 is dimensionless. In order to compensate for some of the boundary layer temperature gradient effects on the various gas properties in rocket combustion, Bartz (Ref. 8-12) has surveyed the agreement between theory and experiment and developed semi-empirical correction factors:

The subscript 0 refers to properties evaluated at the stagnation or combustion temperature; the subscript am refers to properties at the arithmetic mean temperature of the local free-stream static temperature and the wall temperatures; and p is the free-stream value of the local gas density. Again, the empirical constant 0.026 is dimensionless when compatible dimensions are used for the other terms. The gas velocity v is the local free-stream velocity corresponding to the density p. Since density raised to the 0.8 power is roughly proportional to the pressure and the gas film coefficient is roughly proportional to the heat flux, it follows that the heat transfer rate increases approximately linearly with the chamber pressure. These heat transfer equations have been validated for common propellants, limited chamber pressure ranges, and specific injectors (see Ref. 8-13).

The temperature drop across the inner wall and the maximum temperature are reduced if the wall is thin and is made of material of high thermal con

ductivity. The wall thickness is determined from strength considerations and thermal stresses, and some designs have as little as 0.025 in. thickness.

Surface roughness can have a large effect on the film coefficients and thus on the heat flux. Measurements have shown that the heat flow can be increased by a factor of up to 2 by surface roughness and to higher factors when designing turbulence-creating obstructions in the cooling channels. Major surface roughness on the gas side will cause the gas locally to come close to stagnation temperature. However, surface roughness on the liquid coolant side of the wall will enhance turbulence and the absorption of heat by the coolant and reduce wall temperatures.

Example 8-1. The effects of varying the film coefficients on the heat transfer and the wall temperatures are to be explored. The following data are given:

Wall thickness 0.445 mm

Wall material Low-carbon steel

Average conductivity 43.24 W/m2-K/m

Average gas temperature 3033 K or 2760°C Average liquid bulk temperature 311.1 K or 37.8°C

Gas-film coefficient 147 W/m2-°C

Liquid-film coefficient 205,900 W/m2-°C

Vary hg (at constant hi), then vary h/ (at constant hg), and then determine the changes in heat transfer rate and wall temperatures on the liquid and the gas side of the wall.

SOLUTION. Use Eqs. 8-16 to 8-19 and solve for q, Twg, and Twl. The answers shown in Table 8-4 indicate that variations in the gas-film coefficient have a profound influence on the heat transfer rate but relatively little effect on the wall temperature. The exact opposite is true for variations in the liquid-film coefficient; here, changes in hi produce little change in q but a fairly substantial change in the wall temperature.

TABLE 8-4. Change in Film Coefficient Change in

Film Coefficient (%) Change in Wall Temperature (K)

- Heat Transfer -

Gas Film Liquid Film (%) Gas Side, Twg Liquid Side, TK,

TABLE 8-4. Change in Film Coefficient Change in

Film Coefficient (%) Change in Wall Temperature (K)

- Heat Transfer -

Gas Film Liquid Film (%) Gas Side, Twg Liquid Side, TK, 