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To truly understand this we must include the energy equations and turn this graph into Speed of car as a function of required free stream air. This will clearly show us at what point the motor controllers will begin to heat up past a safe point. However solving for car speed, v, in terms of total power, P, is a fucking mess. It is possible but very time consuming. If the reader would like to try they are more than welcome to go ahead but its actually fucked.
v = ((864 A^3 C^3 d^3 (g w sin(a) + g t w)^3 + 2916 A^4 C^4 d^4 P^2)^(1/2) + 54 A^2 C^2 d^2 P)^(1/3)/(3 2^(1/3) A C d) -
(2 2^(1/3) (g w sin(a) + g t w))/((864 A^3 C^3 d^3 (g w sin(a) + g t w)^3 + 2916 A^4 C^4 d^4 P^2)^(1/2) + 54 A^2 C^2 d^2 P)^(1/3)
https://www.desmos.com/calculator/65rszexegd
Heat-Sink Requirements
Since the cold plate will be directly attached to the exterior we need to find the minimum thermal conductivity of the cold plate as well as the minimum air speed to dissipate dissipate the heat through forced convection.
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First we must find the maximum allowable thickness of the cold plate as this will impact if all power created in the motor controller can be dissipated through the plate. The simplified equation of heat flow through a material is governed by the following conduction equation:
Q = (k*A/T)*Δt
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| Description | Value | |
|---|---|---|
| A | Contact area | 0.029 m^2 |
| T | Stack up thickness | TBD |
| k | Thermal Conductivity of the stackup, this value is approximated but should be recalculated with a correct connection | 200 W/m-K |
Since A, Q, and Δt are known we can find the minimum value of k/T.
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A description of the above variables is found in the following table.
| Description | Value to | |
|---|---|---|
| ρ | Density | 1.109 kg/m3 |
| v | Kinematic Viscosity, ρ/μ | 1.750*10-5 m2/s |
| μ | Dynamic Viscosity | 1.941*10-5 kg/m-s |
| k | Thermal Conductivity | 0.0269 W/m-K |
| h | Heat transfer coefficient | 114 W/m^2-K |
| δ | Characteristic Length, for our purposes this will be L | L = 0.117 |
| Cp | Specific heat capacity | 1007 J/Kg-K |
| V | velocity | TBD |
| L | Length of the plate along the direction of flow | 0.117 m |
| Pr | Prandtl Number | 0.7241 |
| A | Area normal to the flow | 0.02925 m^2 |
*Values in the above table are for air at 45C using the following database https://www.engineersedge.com/physics/viscosity_of_air_dynamic_and_kinematic_14483.htm
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