Total Thrust

For cone and plate geometry, the primary normal stress difference T i — 0 2 can be obtained from the vertical thrust force F tending to separate the two surfaces, and this can be measured by a force transducer as in the Weissenberg rheogo-niometer the relation is 

The derivation of this equation assumes that the free surface at the rim of cone and plate has a shape corresponding to part of a sphere of radius / , but experiments with intentionally different surface shapes indicate that for small cone angles the results are insensitive to this feature.  

For parallel plate geometry, the vertical thrust depends on both primary and secondary normal stress differences. A device for providing this measurement at very high shear rates has been described by Walters.  

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Again the reader must be warned that a large proportion of the Je° data in the correlation, summarized in Eqs. (5.26) and (5.27) and Tables 5.2 and 5.5, are based on normal stress measurements (total thrust in plate-cone rheometers) with attendant uncertainties about whether limiting behavior was attained. Also, in... 

The determination of the thrust of a rocket motor involves recording the time diagram of the force (tons, kp, or newtons) during combustion. The force is allowed to act on a support, with a pick-up element thrust cell interposed between them. The measurement is carried out by the aid of a strain gauge element (variation of resistance with pressure) or of a piezo-quartz element, and the results are recorded on an oscillograph connected in a compensation circuit. Modern measuring and computation techniques yield the total thrust time (impulse) immediately. 

However, this is only equal to the total thrust F of the system when the pressure pe at the end of the nozzle is the same as the pressure of its surroundings pa. In general a correction term is needed which is called the pressure term Fpressure. 

If the pressure at the end of the nozzle is smaller than the pressure of the surroundings (which occurs in the case of a so-called overexpanding nozzle), the pressure term then has a negative value and reduces the total thrust. Therefore, a pressure... 

Since the air pressure decreases with increasing flight altitude, at constant nozzle diameter, the total thrust increases with increasing flight altitude. This increase can correspond to approximately 10 to 30 % of the total thrust depending on the rocket. The maximum thrust is reached in vacuo. The so-called effective ejection velocity ceff (of the combustion gases) is defined as the ratio between the thrust and the mass flux (dm / dt) ... 

The so-called total impulse It is the integral of the total thrust integrated over the total combustion time t ... 

Another confirmation that the observed negative behavior represents a true material function is given by Magda al, [32] by comparing negative values derived from total thrust with those derived from the radial normal stress distribution as obtained by small... 

An interesting experimental observation was noted by Magda et al, [32] regarding the effect of smoothness of the rheometer plates. For a solution of flexible polymers, two sets of plates gave identical linear normal stress profiles, as required to extract from a combination of point normal force measurements and total thrust measurements. However for the lyotropic solutions, only the smoother plate yielded a linear profile. They concluded that although it is possible to impose... 

After 5 min with the indenting tip still in position, add the weights to the load-carrying plate so that the total thrust on the test specimen is 50 1N. Apply the extra mass required to increase the load on the specimen to 10 0.2 N (loading 1) or 50 1.0 N (loading 2). 

Equivalent Exhaust Velocity Total thrust divided by the propellant mass flow rate. 

Total axial thrust equals total thrust on the back shroud minus total thrust on the suction ... 

However, this is only equal to the total thrust F of the system when the pressure Pe at the end of the nozzle is the same as the pressure of its surroundings p. In general a correction term is needed which is called the pressure term Fpressure-Figure 2.11 schematically shows the pressure profile which is one factor that influences rocket performance. The length of the arrows shows the contribution of the pressure from inside and outside the walls. While the atmospheric pressure outside is constant, the inside pressure of the combustion chamber is at its largest... 

It is more common and simpler to measure the total thrust on the plate. Making a force balance on the plate, we have... 

Miller and Christiansen (1972) also measured total thrust and obtained the Ni values shown in Figure 5.4.3. By subtracting 2r, (R) from the slopes in Figure 5.4.2, Ni can be calculated independently. Similarly subtracting N] (calculated from the total thrust from the slopes) gives a test of the rim pressure measurements. Results by each of these methods for two cone angles compare well, particularly for N. ... 

This result has been well confirmed experimentally for Newtonian fluids using pressure distribution data (Greensmith and Rivlin, 1953 Markovitz and Brown, 1963 Adams and Lodge, 1964 Alvarez et al., 1985) and from total thrust (Huppler et al., 1967 Miller and Christiansen, 1972 Kulicke, et al., 1977 Whitcomb and Macosko, 1979). Figure 5.4.4 compares eq. 5.4.23 to experimental data. Several of these studies indicate that the correction also seems to be valid for non-Newtonian fluids. 

More frequently, total thrust is measured. Integrating eq. 5.5.10 for gives... 

Normal stress differences for a high density polyethylene (Marlex 6050) at 190°C. Total thrust between parallel disks (solid line)... 

Figure 5.4.3 compares Ni and Wz as determined from total thrust in the cone and plate to values obtained from the pressure distribution. Figure 6.2.13 and Table 6.2.2 compare N from cone and plate to extrudate swell. Figures 6.3.4,6.3.6, and 6.3.7 do the same for N determined by exit pressure and hole pressure. These comparisons helped us to decide which mediods were reliable and over what range of measurement The same ccndd be said of the normal stress comparisons made for axial annular and tangential annular flow in Figure 6.4.2. 

Although nearly all shear normal stress data reported in the literature are measured by total thrust methods, some work has been done with the distribution of pressure across the plate with small flush transducers (Christiansen and Leppard, 1974 Magda et al., 1991 see Figure 5.4.2). Meissner and co-workers (1989) have used the total thrust on a central disk of smaller radius than the sample. Both techniques permit measurement of N2 as well as A i. Birefringence methods, discussed in the next chapter, can also give normal stress distributions. 

Core Energy 15 full power years is a simplifying assumption used for initial concept comparisons. It approximates to 10 years total thrust (full power) time and 10 years reduced power < 50% power) fora 20 year mission. Based on JPL Document 982-00115, Revision 2, Prometheus Project Multi-mission Project Derived Requirements , dated July 15.2005. 

The total thrust force F produced during shear can be related to the normal stress difference in shear. Integrating Equation 8.37 yields PJf) = xjf) -p(r), where P (f) is the integration constant and, physically, is the net pressure exerted by the sheared fluid on a surface with normal vector n. Substituting for p in Equation 8.35 and integrating the resulting equation from rtoR yields... 

The total thrust force F on the lower plate is therefore... 

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