Rocket Science

References & Constants

 
References and Constants used in various aspects of our are placed here. We intend to use this page as our main reference guide and share our finding with the interested readers
 
Electrical Equations used Induction Heating:
P=I^2*R For power consumption of the alloy (From Ohm's Law).
Δ = (2ρm/(μ0μmf))^1/2
Note: 1.		 Assume Solid cylinder, Current penetration depth in cm
Δ= AC Cirrent Penitration depth,
ρ = Metal Resistivity ohm-cm
f = frequency (Hz).
IpNp=IsNs

(Is=IpNp)

 Transformer Current Relationship.(Ns = 1 in metal stock)
Np = Primary Turns
Ns = Secondary Turns (1 T)
Is = Secondary current.
Μ12 = (rs/rp)^2 Coupling Relations between Primary and Secondary windings. (Assumes melt and coil are of the same height)
 Μ12 =  Coupling factor in %
 rp = Primary winding Radius
 rs = Secondary Winding Radius.
       (ID Radius of Crucible or
        OD Radius of cylindrical stock)
Ώm = 2πrsρ/(hΔ) Resistance of Melt using Cylindrical Model
 Ώm = Resistance of melt (Ohms).
h = Cylindrical Height
rs = Cylindrical Radius
Δ  = Cylindrical Current Skin Depth.
Pm = [Ώm][Μ12 ][Is^2]

Pm=[2πrsρ/(hΔ)][(rs/rp)^2][(NpIp)^2]
Note: 2

 Power delivered to the metal melt in terms of all the variables defined above using Cylindrical Model! Q.E.D. 
L = μ0N^2A/l
Note: 3		 Theoretical Solenoid Inductor Eqn:
(Assime length >> radius).
L = Solenoid Inductance    
       (Henries)
A=Area m^2
l = length m
N= Number of turns.
L = (nd)^2/(18d + 40l)
n = (L(18d + 40l))^1/2/d
Note: 4		 Practical solenoid inductor eqn when
(length ~= radius.
L= inductance μH
d = diameter (inches)
l = length (inches)
True for l >= 0.4d
L = [μ0N^2h/2][ln(b/a)] Inductance of a Rectangular Toroid.  The value is heavily dependent on the wire spacing for air core!
L = Inductance in Henries
N = Number of turns
h = height of toroid
b = outer radius
a = inner radius
 
Electrical Equations: Resonant Tanks
f= 1/(2π(LC)^1/2 ) LC Tank Resonant Frequency (f)
L = Tank Inductance (Henries
C = Tank Capacitance (Farads)
z=(L/C)^1/2 LC Impedance Relations (z)
L = Tank Inductance (Henries)
C = Tank Capacitance (Farads)
z=(1/(2πfC) Capacitance required for impedance (z).  Then use the other eqn to solve for L
f = Frequency (Hertz)
C = Capacitance (Farads)
Ipk = √2Irms Peak crrent( Ipk ) relationship to RMS current.  Assumed Sinusoidal wave.
Vpk = √2Vrms Peak voltage ( Vpk ) relationship to RMS voltage.  Assumed Sinusoidal wave.
V=IR Ohm's Law: Current (I) through resistor (R) has V volts across R.
R= Resistance (Ohms)
I = Current (Amps) 
P=V^2/R Power Relations with Voltage and Resistance (or Impedance).
Prms=(Vpk)^2/(2R) RMS Power relations with Vpk
Vpk = Peak Voltage
R    =  Resistance (Ohms).
R = (Vpk)^2/(2Prms) Resistance Required to get the power (Prms) and applied volatage (Vpk).
Vpk = Low-side applied voltage.
 Prms = Resonant Tank RMS power.
C = (2Prms)/[2πf(Vpk)^2]

L = 1/[(2πf)^2C]
Note:5

 Tank Capacitance computed from frequency (f), power (Prms), and applied voltage (Vpk).
Q.E.D
 
Electical  Constants usind in our projects
Note: 6
μ0 = 1.26*10-6 H/m
μ0 = 1.26*10-8 H/cm		 Electical Permittivity used for magnetic and 'skin currents'
ρcopper  = Electrical Resistivity of Copper.
Ohm-cm
ρnickel  = Electrical Resistivity of Nickel
Ohm-cm.
ρinco718  = Electrical Resistivity of
INCO-718. Ohm-cm.
ρHaynes242  = Electical Resistivity of Haynes Alloy-242. Ohm-cm
ρr41  = Electrical Resistivity of Haynes
R-41 (Rene-41). Ohm-cm
ρHastalloy-X  = Electrical Resistivity of
Hastalloy-X. Ohm-cm
 
 
Chemistry
PV=nRT Universal Gas Law. We use this to measure contaminants in the vacuum chamber.  We also use this in the thermodynamics of the rocket engines.
P = Pressure.
V = Volume
T = Temperature.
n(N) = Moles or (gas number)
R= Universal Gas Constant
      (Comes in many forms).
P1V1/T1 = P2V2/T2 Another important gas relations we use often. This works for Isentropic expansion as well.
Gas Constant Table Important link to the gas constant tables using different units.
6.02 * 10^23 Avagado's Number
e- = 1.6 * 10-19
 		 Electron Charge used in electo-forming processes, electron beam welding.
Charge of electron in Coulomb.
1 A = 1 Coulumb/sec
= 6.25 * 10+18 electrons/Amp
= 1.04 * 10-5 Moles e/Amp
Note: 7		  
γ = Cp/Cv Specific Heat Ratui (γ) used to analyse many thermodynamic systems involving gases.
Cp = Specific Heat at constant
         pressure.
Cv = Specific Heat at constant
       volume.
 
Material Science
ΔT = PΘ  Θ = Thermal resistance of the
         matterial in Kelvin/Watt.
P = power in watts.
ΔT = Temperature across
         material.
   
 
Material Science Constants:
Θ =  
   
   

 
Notes:

1. Current flow at skin depth dependent on frequency.  This equation applied to the solid cylinder of metal inside a primary coil.  The cylinder behaves as a shorted single turn secondary. This equation is used to determin the depth of the current to establish the effective current geometry of the current flow.  With the current flow model, I ^2R heating power can be computed.

2. Power absorbed as heat in the metal cylinder (I ^2R). The equation takes into account mutual coupling effects (M) and the effective curremt path geometry (Current flowing tangentailly on the exterior cylinder wall of resistivity ρ.  Finally, the equation computes the power absorbed in terms of the primary winding current. Typically the primary current is between 240A to 350 A with the primary winding of 12 to 16 turns.  Mutual inductance plays an important role in the coupling or transfer of power between the primary and the melt. IMPORTANT: In our project, this model applies to our circuit which work under current mode, this means that the coupling and power transfer is true only if the theoretical secondary voltage ratioed  (by the turns ratio) is less than the theoretical primary voltage.  That is Vs << [M][Vp][(Ns/Np)].  This holds true only if the circuit is designed to run in current mode.  Note if the theoretical secondary voltage fails to meet the above relations, one or both of the following will occur:
    A.  Power transfer will drop.
    B.  Resonant tank power overload can occur and destroy the
          electronic componentsl

3. The equation to determine the inductance (L in Henries) of an ideal solenoid inductor.  This equation is only accurate for cases where the length is significantly longer than the diameter of the solenoid  (l>>d).

4. This equation is a practical and accurate estimation of inductance (L in micro-henry) for solenoid inductors for which the length is 40% of the diameter or longer (l>= 0.4d).  It only applies to a single winding layer of an air core.

5. The set of equations used to determine the inductance (Henries) and capacitance (Farads) of a parallel resonant LC tank to deliver magnetic power density in the primary core.  This mode assumes the transformers and inductors are operating in context of our project.

6. The constants in this section is for our reference.  It contains constants such as resistivity (ρ) of different metal alloys we are using in the manufacturing processes (e.g. induction melting).  Many of these constants are temperature dependent.

7. These are variations of the amount of electron flow per Ampere of electrical current.  The other constants are adaptations for electro-forming (electroplating processes).  We intend to use electroforming to form thin shelled but strong components.  For instance, after casting a rocket engine chamber, electroplating a Nickel-Cobalt alloy cooling jacket/passage closureand heat treat the finished product will complete the structure of the component.

 
 
By the Rocket Scientists at home.
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Revised: 11/18/03.