DDW Idealized Propeller Cart Calculator (ALPHA)

"When the fields are green, the data is clean!"

Every Compute button press recalculates all parameters.
Entering just a number without units uses the last active units.
Entering just units without a number converts the value to those units.
When a value is changed, it's text background color is set to light blue.
Hover over parameter input text box for popup tooltip showing available units.
Many parameters have two flavors; the first has units and the second is dimensionless.
Dimensionless velocity parameters are normalized with respect to Vwind.
Dimensionless force parameters are normalized with respect to Fwind.
Dimensionless power parameters are normalized with respect to Pwind.
EQ16 determines the Δ value that maximizes Fnet and Pnet for a given n.
Animations run in one of three propeller modes: fixed pitch, fixed ΔV, or optimal ΔV.
You can enter commands into any text box having form: COMMAND=PARAMETER. See below.
You can change the units of all parameters - e.g. UNITS=METRIC, UNITS=IMPERIAL (or U=SI, U=IMP).
You can change units for all parameters having common units - e.g. FORCE=N, SPEED=m/s.
You can multiply or divide a value by a number - e.g. *4, /3.14.
You can enter a rational number using integers - e.g. 4/3, 16/27.

Equations used:

EQ1: Pwind = 1/2 * ρ * S * Vwind3
EQ2: Fwind = 1/2 * ρ * S * Vwind2
EQ3: n = Vcart / Vwind
EQ4: Δ = (V2 - V1) / Vwind
EQ5: V1 = Vcart - Vwind
EQ6: V2 = V1 + ΔV
EQ7: V = (V1 + V2) / 2
EQ8: A1 = S * V / V1
EQ9: A2 = S * V / V2
EQ10: MDOT = ρ * S * V
EQ11: Pprop = 1/2 * ρ * S * Vwind3 * ( 1/2*Δ3 + (2*n - 2)*Δ2 + (2*n2 - 4*n + 2)*Δ )
EQ12: Ft = 1/2 * ρ * S * Vwind2 * ( Δ2 + (2*n - 2)*Δ )
EQ13: Fg = 1/2 * ρ * S * Vwind2 * (1/2*Δ3 + (2*n - 2)*Δ2 + (2*n2 - 4*n + 2)*Δ) / n
EQ14: Fnet = 1/2 * ρ * S * Vwind2 * (-1/2*Δ3 + (2 - n)*Δ2 + (2*n - 2)*Δ ) / n
EQ15: Pnet = 1/2 * ρ * S * Vwind3 * ( -1/2*Δ3 + (2 - n)*Δ2 + (2*n - 2)*Δ )
EQ16: Δmax = 2/3 * (2 - n + (n2 - n + 1)1/2)
EQ17: VSR = Pit * Tw / (PI * Dw * Tr)
EQ18: nmax = 1 / (1 - VSR)


Description of Dimensionless parameters:

n = Velocity of cart (IFOR=gnd) = Vcart/Vwind
Δ = Change of air velocity (IFOR=any) = ΔV/Vwind = o - i
a = "Axial Induction Factor" 1/2 Change of air velocity (IFOR=any) = Δ / 2
i = Velocity of air entering control volume inlet (IFOR=cart) = V1/Vwind = n - 1 = o - Δ
o = Velocity of air exiting control volume outlet (IFOR=cart) = V2/Vwind = i + Δ
v = Velocity of air passing through rotor disc (IFOR=cart) = V/Vwind = i + Δ/2 = o - Δ/2
a1 = Area of control volume inlet = A1/S
a2 = Area of control volume outlet = A2/S
CPprop = Coefficient of power passing from propeller to air (IFOR=cart) = Pprop/Pwind
CFt = Coefficient of thrusting force of air acting on propeller = Ft/Fwind
CFg = Coefficient of braking force of ground acting on rear wheel = Fg/Fwind
CFnet = Coefficient of net force = Fnet/Fwind = (Ft - Fg)/Fwind = CFt - Cfg
CPnet = Coefficient of net power (IFOR=cart) = Pnet/Pwind = CFnet * n
VSR = "Vehicle Speed Ratio" speed of prop through the air at the disk / speed of cart along ground = V / Vcart
VSR is also the distance through the air / distance along the ground = Pit * Tw / (PI * Dw * Tr)
nmax = "Pitch Speed" Maximum fixed pitch cart velocity (where Δ => 0) = 1 / (1 - VSR)


Input Parameters

Environment:	Wind speed:	Vwind =		Air density:	ρ =
Propeller:	Diameter:	Dr =		Rotor disc Area:	S =
Mode:	Fixed pitch	Pit =		CP radial station:	CRcp =
	Fixed ΔV	Δ =			a =
	Optimal ΔV	Δmax =			amax =
Wheel:	Diameter:	Dw =		Circumference:	Cw =
Transmission:	Wheel axle gear teeth:	Tw =		Rotor shaft gear teeth:	Tr =

Misc / Rotor Kinematics

Maximum fixed pitch speed:	nmax =		->	->	->	->
Vehicle speed ratio (V / Vcart) :	VSR =

Nondimensionalization Reference Constants

Wind Power (IFOR=ground):	Pwind =			->	->	->
Wind Force (IFOR=ground):	Fwind =			->	->	->

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Primary computed parameters

Cart velocity (IFOR=ground):	Vcart =		= n
Air velocity change due to rotor:	ΔV =
Inlet air velocity (IFOR=cart):	V1 =		= i		Lower n =
Outlet air velocity (IFOR=cart):	V2 =		= o		Upper n =
Rotor disk air velocity (IFOR=cart):	V =		= v		n increment =
Area of control volume inlet:	A1 =		= a1		Timing interval =
Area of control volume outlet:	A2 =		= a2
Mass flow rate (IFOR=cart):	MDOT =			->	->
Pin=(Fg * Vcart) = Pout=(Ft * V):	Pprop =		= CPprop	->	->	->
Thrust force of air acting on rotor:	Ft =		= CFt	->	->	->
Force of ground acting on wheel:	Fg =		= CFg	->	->	->
Net force (Ft - Fg):	Fnet =		= CFnet	->	->	->
Net Power (Fnet * Vcart):	Pnet =		= CPnet	->	->	->

Secondary computed parameters

Wheel axle torque (Fg * Dw/2):	Qw =		Wheel axle rotational speed (Vcart / (Dw/2)):	ωw =
Rotor shaft torque (Qw * Tr / Tw):	Qr =		Rotor shaft rotational speed (ωw * Tw / Tr):	ωr =
Advance ratio (V1 / (ωr * Dr/2)):	J =		Tip speed ratio (1 / J):	TSR =
Rotor tip radial speed (ωr * Dr/2):	Vqtip =		Rotor tip apparent wind angle (ATAN(V / Vqtip)):	ϕ =
Rotor CP radial speed (ωr * CRcp * Dr/2):	Vqcp =		Rotor CP apparent wind angle (ATAN(V / Vqcp)):	ϕcp =