Richard Nakka’s Experimental Rocketry Web Site

Solid Rocket Motor Theory – Two-phase Flow

Example 1

For KNSU propellant, calculate the various two-phase thermochemical and performance parameters:

· Molecular mass, specific gas constant and ratio of specific heats

· Critical (throat) velocity, Characteristic Exhaust Velocity

· Exhaust velocity and Specific Impulse (ideal expansion from 1000 psi chamber pressure)

The first step is to run ProPep to obtain the products of combustion:

Code WEIGHT D-H DENS COMPOSITION

0 POTASSIUM NITRATE 65.000 -1169 0.07620 1 N 3 O 1 K

0 SUCROSE (TABLE SUGAR) 35.000 -1550 0.05740 22 H 12 C 11 O

THE PROPELLANT DENSITY IS 0.06836 LB/CU-IN OR 1.8923 GM/CC

THE TOTAL PROPELLANT WEIGHT IS 100.0000 GRAMS

NUMBER OF GRAM ATOMS OF EACH ELEMENT PRESENT IN INGREDIENTS

2.249436 H

1.226965 C

0.642877 N

3.053349 O

0.642877 K

****************************CHAMBER RESULTS FOLLOW *****************************

T(K) T(F) P(ATM) P(PSI) ENTHALPY ENTROPY CP/CV GAS RT/V

1720 2636 68.02 1000.00 -130.24 165.92 1.1331 2.382 28.559

SPECIFIC HEAT (MOLAR) OF GAS AND TOTAL = 10.505 14.984

NUMBER MOLS GAS AND CONDENSED = 2.382 0.307

7.969284e-001 H2O 5.322495e-001 CO 3.880990e-001 CO2 3.213577e-001 N2

3.134432e-001 H2 3.065426e-001 K2CO3* 2.796753e-002 KHO 1.473071e-003 K

1.537422e-004 K2H2O2 1.005049e-004 NH3 1.856416e-005 H 1.306551e-005 KH

1.042846e-005 KCN 6.919009e-006 CH4 3.484479e-006 CH2O 3.326461e-006 CNH

2.62602E-06 HO

The next step is to calculate the number of moles, mole fraction, mass and mass fraction for each of the major products of combustion. Trace products such as K, K2H2O2, NH3 will be neglected. The products are separated in terms of condensed phase and gaseous. For KNSU, the sole condensed phase product is potassium carbonate (K2CO3). ProPep indicates liquid products with an asterisk (*) and solids with an ampersand (&). The calculations are most conveniently done using a spreadsheet app such as Excel. The results for KNSU are shown in Table 1.

Table 1

The molecular mass (MW) for the products may be found at Wikipedia or NIST Chemical Webbook.

Using nitrogen, gas N2, as an example, the mole fraction, mass and mass fraction are calculated as such:

· mole fraction N2 = # moles N2 / # moles gas = 0.3214/2.380 = 0.1350

· mass N2 = MW N2 × # moles N2 = 28.02 ×0.3214 = 9.004 grams

· N2 mass fraction of gases = mass N2/mass gas = 9.004 / 57.556 = 0.1560

· N2 mass fraction of mixture= mass N2/mass mixture = 9.004 / 100.0 = 0.0900

The mass fraction of condensed phase, X, is calculated as:

X = mass condensed / total mass of products = 42.367 / 100 = 0.424

Effective molecular mass, Mʺ, is calculated as:

Mʺ = total product mass / number of gas moles = 100.0 / 2.38 = 42.02 grams/mole

Specific gas constant, Rʺ, is calculated as the universal gas constant divided by the effective molecular mass: (where = 8314 J/kmol-K)

Rʺ = / 42.02 = 197.9 J/kg-K

The specific heat of the condensed phase products, Cs, and the gaseous products, Cp, is found using Eqn.4a:

Both Cs and Cp are functions of temperature. As such, we will determine the values of the specific heat of the products at chamber conditions (To) and assume a “frozen-flow” model whereby the products of combustion do not change as the products flow through the nozzle. Alternatively, the average temperature of the products from chamber conditions to nozzle exit conditions can be used, however, the simplification of using chamber temperature results in a small overall difference.

The specific heat for each of the products at the combustion temperature is calculated using the Shomate polynomial equation:

Cp = A + B*t + C*t² + D*t³ + E/t²

where Cp = heat capacity (J/mol*K) and t = temperature (K) / 1000. The values for the polynomial coefficients can be found at the NIST Chemical Webbook.

Using nitrogen gas (N2) at KNSU combustion temperature of 1720 K. as an example:

Valid for temperature range 500K – 2000K

A 19.50583

B 19.88705

C -8.598535

D 1.369784

E 0.527601

This gives, for N2 at 1720 K:

35.42 J/mol-K

Once again, using Excel greatly eases the calculations and minimizes potential for calculation error. The results for all the KNSU products are shown in Table 2.

Table 2

Note that the units applicable to the Shomate equation are Joules per mole-Kelvin (J/mol-K). We want the heat capacity to be per unit mass. In the second row in Table 2, the units of J/gram-K are arrived at by dividing the molar Cp by the molecular mass of each product. The Cp for the gas-only mixture and the Cp for the gas-condensed mixture is given by Eqn.4a:

Note that m_i/m_p (i=1,2,3…) is the mass fraction of the particular product as calculated in Table 1. This results in the following:

The unit of measure we need to use for the thermochemical calculations is Joules per kilogram-Kelvin (J/kg-K) and as such, the values we need for Cp_(gas), Cp_mix and Cs are each multiplied by 1000, giving:

Cp _(gas) = 1813.4 J/kg-K

Cp_mix = 1685.0 J/kg-K

Cs = 1513.6 J/kg-K

The specific heat ratio for the mixture is calculated using Eqn.3