Checking the solid example’s peak-pressure numbers by hand

The numbers

Open File → Open Example → Solid rocket and press Solve. On Review, the table “At peak chamber pressure” lists these among others (version 2.0 in US units; the SI column is converted for this page):

The propellant is KNSB, potassium nitrate and sorbitol 65/35, with the burn-rate law and the ideal density of 1,841 kg/m³ that Richard Nakka published. Define → Propellant shows the source next to each value.

1. Mass flow from c*

The characteristic velocity is defined by c* = p_c A_t / ṁ (Sutton and Biblarz, chapter 3). The throat area is π/4 × 0.3740² = 0.1099 in², so p_c A_t = 512.7 × 0.1099 = 56.32 lbf. In US units ṁ = p_c A_t g_c / c* = 56.32 × 32.174 / 2980 = 0.6081 lb/s, against 0.6082 on the page. The last digit differs because c* is shown rounded to 2980 ft/s.

2. Burning area and density from Kn and the burn rate

Kn is the burning surface area divided by the throat area, so at this instant the grain burns on A_b = 273.7 × 0.1099 = 30.07 in². In a quasi-steady burn the gas leaving through the throat is the propellant burning, ṁ = ρ_p A_b r. Solved for the density, ρ_p = 0.6082 / (30.07 × 0.3041) = 0.06652 lb/in³, which is 1,841 kg/m³, the density in the propellant entry.

3. The burn rate from Nakka’s law

Nakka gives KNSB’s burn rate as r = a pⁿ in pressure ranges, with r in mm/s and p in MPa (strand-burner measurements, his table 2). 512.7 psi is 3.535 MPa absolute, inside the range from 1.50 to 3.79 MPa, where a = 7.852 and n = −0.013. Then r = 7.852 × 3.535^−0.013 = 7.724 mm/s, or 0.3041 in/s.

4. The chamber pressure from the other three

The first two relations together give p_c = ρ_p c* Kn r. In SI units, 1841 × 908.3 × 273.7 × 0.007724 = 3.535 MPa, the 512.7 psi on the page. With r = a pⁿ this becomes the steady chamber pressure of a solid motor, (p_c/p_u)¹⁻ⁿ = Kn ρ_p a c* / p_u, where p_u is the pressure unit of the law (1 MPa here) and a is converted to m/s. This is the balance the app solves at each point of the burnback (Sutton and Biblarz, chapter 12), with the gas stored in the volume freed by the burnt propellant neglected. With n this close to zero, p_c varies as Kn^0.987: almost in proportion to Kn.

Using the same checks on your own motor

If the numbers do not close for a motor of your own, the usual causes are a burn-rate coefficient entered on the wrong unit basis (mm/s with MPa against in/s with psi; the app stores the basis with each range), a density or c* typed in the wrong units, or a range boundary: where a law’s ranges do not meet, the app can hold the pressure at a range’s bound, and the burn rate then follows from the mass balance rather than from the law.