Richard Nakka's Experimental Rocketry Web Site


Rocket Motor Design Charts
- Chamber Pressure -

  • Introduction
  • Design Charts
  • Development of Charts
  • Determination of Kn
  • Example of Usage

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    Introduction

    The web page presents design charts which may be used to determine the steady-state 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: For the chart data to be considered valid, it is necessary that the propellant be prepared by the "standard" method: The term steady-state infers the operating condition whereby chamber pressure is solely a function of grain burning-surface 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 build-up as well as the pressure tail-off 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 cross-sectional area) in order to utilize the charts, the methodology for calculation of surface area for a hollow-cylindrical grain and BATES grain are presented. Examples of such calculations and chart usage are also provided.


    Design Charts

    Sorbitol

    Figure 1-- Design chart for Sorbitol based propellant.


    Dextrose

    Figure 2-- Design chart for Dextrose based propellant.


    Sucrose

    Figure 3-- Design chart for Sucrose based propellant.

    Note on rev 2009-04-01 of charts: The charts were revised to use the correct value of the specific heat ratio and a reduced c-star 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 steady-state chamber conditions:
    equation 1

    where the parameters are defined as:
    Po   motor chamber pressure
    Kn   Klemmung, Kn=Ab /At (Ratio of Burning surface area (Ab) to the nozzle throat cross-section area (At) )
    a   Burn rate pressure coefficient
    a   Burn rate pressure conversion factor, MPa to Pa units (a =1 000 000n )
    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 KN-Dextrose & KN-Sorbitol 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 1-3:

    c-star

    Data table

    Table 1

    Data table

    Table 2

    Data table

    Table 3


    Determination of Kn

    grain

    Figure 4-- Hollow cylindrical grain

    The burning surface area for a hollow-cylindrical grain, as shown in Figure 4, may be calculated as follows:

    Grain with unrestricted burning (no surfaces inhibited):
    Ab max = Ab initial = ½ p (D 2 - d 2) + p L (D + d)
    Ab final = p (D + d) (L - t)       where t = ½ (D - d)
    Grain with outer surface inhibited (burning at core and ends):
    Ab max = Ab initial = ½ p (D 2 - d 2) + p d L
    Ab final = p D (L - 2t)
    Grain with both ends inhibited (burning on outer surface and core):
    Ab = constant = p L (D + d)

    grain

    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 one-half the web thickness, to produce a symmetric profile (initial Kn = final Kn).

    grain

    Figure 6-- BATES Kn profiles

    The instantaneous grain burning surface area is given by:

    Ab = N [½ p (D2 - d2) + 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.

    BATES

    Figure 7-- BATES grain illustrating regression of the web

    The initial and final burning surface areas are given by:
    Ab initial = N [½ p (D2 - do2) + p Lo do]
    Ab 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. dAb/dx = 0), then solving for x.

    As such, the value of x is found to be:
    x = 1/6 (Lo - 2do)        at Ab max
    The value of Ab 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, Ab max = Ab final
    The Kn profile is regressive if the calculation gives   x < 0. In this case, Ab max = Ab 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.


    rod & tube grain

    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:

    Ab =p (d + Dr) 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:

    The gap width should be chosen such that the initial port area (Ap) is a minimum of 2 times the nozzle throat cross-sectional area (At).

    Ap > 2 At    or

    p /4 (d2 - Dr2) > 2p /4 Dt2     (where Dt is throat diameter). or,

    (d2 - Dr2) > 2 Dt2


    Example of Usage

    Example 1
    Determine the initial, maximum, and final steady-state 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:
    Ab max = Ab initial = ½ p (D 2 - d 2) + p L (D + d)
    This gives        Ab max = Ab initial = ½ p (2.25 2 - 1.00 2) + p 10.50 (2.25 + 1.00) = 114 in2

    Ab final = ½ p (D + d) (L - t)       where t = ½ (D - d)
    Therefore        t = ½ (2.25 - 1.00) = 0.625 in.
    and       Ab final = p (2.25 + 1.00) (10.50 - 0.625) = 101 in2

    The nozzle throat cross-sectional area is:    At = ¼ p (0.650)2 = 0.332 in2
    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 steady-state chamber pressure is found to be 1080 psi. The final steady-state chamber pressure is 950 psi.


    Example 2
    Determine the initial, maximum, and final steady-state 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:       Ab = N [½ p (D2 - d2) + p L d]
    Giving      Ab max = 3 [½ p (752 - 40.72) + p (81.3) 40.7] = 49 890 mm2
    The initial and final burning surface areas are given by:
    Ab initial = N [½ p (D2 - do2) + p Lo do]
    Ab final = N p D (Lo - 2t)      where t = ½ (D - do)

    This gives:
    Ab initial = 3 [½ p (752 - 222) + p (100) 22] = 44 960 mm2
    Initial web thickness is      t = ½ (75 - 22) = 26.5 mm
    And       Ab final = 3 p 75 [100 - 2(26.5)] = 33 220 mm2
    The nozzle throat cross-sectional area is: At = ¼ p (13)2 = 133 mm2
    The initial, maximum and final Kn can now be calculated:
    Kninitial = 44 960 / 133 = 338
    Knmax = 49 890 / 133= 375
    Knfinal = 33 220 / 133= 250

    From Figure 2, the maximum steady-state chamber pressure is found to be 6.3 MPa. The initial and final steady-state 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?
    Solution:
    Lo = ½ (3D + do)
    This gives   Lo = ½ [3(50) + 18) = 84 mm.

    Verify that initial and final surface areas are identical:
    Ab initial = 4 [½ p (502 - 182) + p (84) 18] = 32 673 mm2
    Ab final = 4 p 50 [84 - 2(16)] = 32 673 mm2      where t = ½ (50 - 18) = 16 mm


    Last updated

    Originally posted  May 18, 2001
    Last updated March 5, 2016


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