Richard Nakka's Experimental Rocketry Web Site
Rocket Motor Design Charts
 Chamber Pressure 
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Introduction
The web page presents design charts which may be used to determine the steadystate chamber pressure of a solid rocket motor. A "chamber pressure v.s. Kn chart" is provided for each of the following three potassium nitrate based propellants:
 Sorbitolbased KNSB propellant
 Dextrosebased KNDX propellant
 Sucrosebased KNSU propellant
For the chart data to be considered valid, it is necessary that the propellant be prepared by the "standard" method:
 the propellant must be heat cast
 the oxidizer must be finely ground such that the maximum particle size is 75100 micron (e.g. ground by an electric coffee grinder)
 the constituents must be very well blended prior to casting (e.g. 1 hour per 100 g. in a rotating mixer)
 be of the standard 65/35 O/F ratio
 the mass density ratio of the cast propellant should be in the range of 9498% of theoretical
The term steadystate infers the operating condition whereby chamber pressure is solely a function of grain burningsurface area. In other words, the generation of combustion gases, and outflow of gases through the nozzle, are in a state of equilibrium (balance). Therefore, this excludes the initial pressure buildup as well as the pressure tailoff at burnout.The equations and data that were used to develop the design charts are also presented.
As it is necessary to know Kn of the motor (the ratio of surface burning area to nozzle throat crosssectional area) in order to utilize the charts, the methodology for calculation of surface area for a hollowcylindrical grain and BATES grain are presented. Examples of such calculations and chart usage are also provided.
Design Charts
Figure 1 Design chart for Sorbitol based propellant.
Figure 2 Design chart for Dextrose based propellant.
Figure 3 Design chart for Sucrose based
propellant.
Note on rev 20090401 of charts: The charts were revised to use the correct value of the specific heat ratio and a reduced cstar to reflect an assumed combustion efficiency of 95%. The overall difference compared to the earlier charts is quite small, being less than 5% pressure difference for a given kn.
Development of Charts
The three design charts were constructed by use of the following expression for steadystate chamber conditions:
 where the parameters are defined as:

Po motor chamber pressure
Kn Klemmung, Kn=A_{b} /A_{t} (Ratio of Burning surface area (A_{b}) to the nozzle throat crosssection area (A_{t}) )
a Burn rate pressure coefficient
a Burn rate pressure conversion factor, MPa to Pa units (a =1 000 000^{n} )
r propellant mass density
c* propellant characteristic exhaust velocity
n Burn rate pressure exponent
The burn rate pressure coefficients and pressure exponents are based on experimental Strand Burner measurements. Details are provided in the KNDextrose & KNSorbitol Propellants  Burn Rate Experimentation web page. For KNSU, the values used were a=0.0665 in/sec (8.26 mm/sec) and n=0.319, with these values obtained from an earlier series of Strand Burner measurements.
The characteristic exhaust velocity is calculated as shown below. The applicable parameters for each propellant type are given in Tables 13:
Table 1
Table 2
Table 3
Determination of Kn
Figure 4 Hollow cylindrical grain
The burning surface area for a hollowcylindrical grain, as shown in Figure 4, may be calculated as follows:
 Grain with unrestricted burning (no surfaces inhibited):

A_{b max} = A_{b initial} = ½ p (D ^{2}  d ^{2}) + p L (D + d)

A_{b final} = p (D + d) (L  t) where t = ½ (D  d)
Grain with outer surface inhibited (burning at core and ends):
A_{b max} = A_{b initial} = ½ p (D ^{2}  d ^{2}) + p d L
A_{b final} = p D (L  2t)
Grain with both ends inhibited (burning on outer surface and core):
A_{b} = constant = p L (D + d)
Figure 5 BATES grain
A BATES grain configuration is shown in Figure 5. This usually consists of two or more propellant segments, inhibited on the outer surfaces. This configuration is typically used when a nearly neutral Kn profile is desired (red curve in Figure 6). Kn slope rises to a maximum value then decays. The shape of the curve is determined by the Lo/D and do/D ratios. Judicious choice of segment length and core diameter is necessary, or else the Kn profile may instead have only a progressive profile (green curve) or regressive profile (blue curve). Ideally, Knmax should occur at a surface regression of onehalf the web thickness, to produce a symmetric profile (initial Kn = final Kn).
Figure 6 BATES Kn profiles
The instantaneous grain burning surface area is given by:
A_{b} = N [½ p (D^{2}  d^{2}) + p L d] (Eqn.1)
where N is the number of segments; d and L are the instantaneous values of core diameter and segment length, and are given by:
d = do + 2x and L = Lo  2x (Eqns.2a, 2b)
where x is the linear surface regression (distance the web has burned, normal to the web surface). This is illustrated in Figure 7. The dashed lines represent the geometry of the burning surfaces at some arbitrary point of surface regression.
Figure 7 BATES grain illustrating regression of the web
The initial and final burning surface areas are given by:
A_{b initial} = N [½ p (D^{2}  do^{2}) + p Lo do]
A_{b final} = N p D (Lo  2t) where t = ½ (D  do)
The value of x when the burning surface area reaches maximum is important as this determines maximum chamber pressure. This value of x may be found by setting the derivative (represented by the slope of the Kn v.s. web regression curve) to zero (i.e. dA_{b}/dx = 0), then solving for x.
 As such, the value of x is found to be:
 x = 1/6 (Lo  2do) at A_{b max}
The value of A_{b max} is then found by substituting x into Eqns. 2A and 2B to find d and L, then substituting these values into Eqn. 1.
Note that the Kn profile is progressive if the calculation gives x > do. In this case, A_{b max} = A_{b final}
The Kn profile is regressive if the calculation gives x < 0. In this case, A_{b max} = A_{b initial}
 When designing a rocket motor, the dimension D is usually limited by factors such as casing or fuselage size. The choice of core diameter, do, is usually based upon desired web thickness (which determines burn time) and erosive burning considerations. Thus, segment length, Lo, is the parameter that may be available to control the Kn profile. The value of Lo may be found which gives a symmetric Kn profile (initial and final Kn are equal), if D and do are specified:
 Lo = ½ (3D + do) for symmetric profile
The "flatness" of the curve, which is always concave downward, is dependant upon the do/D ratio. As do/D approaches unity, the concavity approaches a straight line.
Figure 8 Rod & Tube grain
The burning surface area for a Rod & Tube, as shown in Figure 8, is completely neutral (does not change during the burn) and may be calculated as follows:
A_{b} =p (d + D_{r}) L
However, the diameter of the Rod Grain needs to be equal to twice the web thickness of the Tube Grain in order for this to be strictly true. This ensures that burnout of the Rod Grain occurs simultaneously to that of the Tube Grain.
Dr = (Dg  d)/ 2
As well, the following surfaces are inhibited from burning:
 Tube Grain outer surface
 Tube Grain ends (both)
 Rod Grain ends (both)
The gap width should be chosen such that the initial port area (Ap) is a minimum of 2 times the nozzle throat crosssectional area (At).
Ap > 2 At or
p /4 (d^{2}  Dr^{2}) > 2p /4 Dt^{2} (where Dt is throat diameter).
or,
(d^{2}  Dr^{2}) > 2 Dt^{2}
Example of Usage
 Example 1

Determine the initial, maximum, and final steadystate chamber pressure for an unrestricted hollow cylindrical grain of KNDX with the following dimensions:
 Outer diameter 2.25 inches
 Core diameter 1.00 inches
 Grain length 10.50 inches
The nozzle throat diameter is 0.650 inches.Solution:
A_{b max} = A_{b initial} = ½ p (D ^{2}  d ^{2}) + p L (D + d)
This gives A_{b max} = A_{b initial} = ½ p (2.25 ^{2}  1.00 ^{2}) + p 10.50 (2.25 + 1.00) = 114 in^{2}

A_{b final} = ½ p (D + d) (L  t) where t = ½ (D  d)
Therefore t = ½ (2.25  1.00) = 0.625 in.
and A_{b final} = p (2.25 + 1.00) (10.50  0.625) = 101 in^{2}
The nozzle throat crosssectional area is: At = ¼ p (0.650)^{2} = 0.332 in^{2}
This gives an initial, and maximum, Kn = 114 / 0.332 = 343. The final Kn = 101 / 0.332 = 304.
From Figure 2, the initial and maximum steadystate chamber pressure is found to be 1080 psi. The final steadystate chamber pressure is 950 psi.
 Example 2

Determine the initial, maximum, and final steadystate chamber pressure for a BATES grain configuration of KNSB with the following dimensions:
 Outer diameter 75 mm.
 Core diameter 22 mm.
 Segment length 100 mm
 3 segments
The nozzle throat diameter is 13 mm.Solution:
 The value of web regression, x, at the point of maximum chamber pressure is found from the following expression:
 x = 1/6 (Lo  2do)
Therefore, x = 1/6 [100  2(22)] = 9.33 mm
Substitute the value of x into the following equations:
d = do + 2x and L = Lo  2x
Giving d = 22 + 2(9.33) =40.7 mm and L = 100  2(9.33) = 81.3 mm
Substitute the values of D, L and d into the equation for burning area: A_{b} = N [½ p (D^{2}  d^{2}) + p L d]
Giving A_{b max} = 3 [½ p (75^{2}  40.7^{2}) + p (81.3) 40.7] = 49 890 mm^{2}

The initial and final burning surface areas are given by:
A_{b initial} = N [½ p (D^{2}  do^{2}) + p Lo do]
A_{b final} = N p D (Lo  2t) where t = ½ (D  do)
This gives:
A_{b initial} = 3 [½ p (75^{2}  22^{2}) + p (100) 22] = 44 960 mm^{2}
Initial web thickness is t = ½ (75  22) = 26.5 mm
And A_{b final} = 3 p 75 [100  2(26.5)] = 33 220 mm^{2}
The nozzle throat crosssectional area is: At = ¼ p (13)^{2} = 133 mm^{2}
The initial, maximum and final Kn can now be calculated:
Kn_{initial} = 44 960 / 133 = 338
Kn_{max} = 49 890 / 133= 375
Kn_{final} = 33 220 / 133= 250
From Figure 2, the maximum steadystate chamber pressure is found to be 6.3 MPa. The initial and final steadystate chamber pressure is 5.0 MPa and 3.1 MPa, respectively.
Example 3
A rocket motor with a BATES grain configuration of KNDX is to be designed to give an approximately neutral burn profile, with matching initial and final Kn values (producing the maximum Kn midway through the web regression). For the following segment and core diameters, what length should the segments initially be?
 Outer diameter 50 mm.
 Core diameter 18 mm.
 4 segments
 Solution:

Lo = ½ (3D + do)
This gives Lo = ½ [3(50) + 18) = 84 mm.
Verify that initial and final surface areas are identical:
A_{b initial} = 4 [½ p (50^{2}  18^{2}) + p (84) 18] = 32 673 mm^{2}
A_{b final} = 4 p 50 [84  2(16)] = 32 673 mm^{2} where t = ½ (50  18) = 16 mm
Originally posted May 18, 2001
Last updated March 5, 2016
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