Motor Controller Heat Dissipation

*Note this page is a pretty wacky, I only recommend reading if you really really care. It gets relatively in depth on things that dont entirely matter and not all the logic is fully documented. I recommend asking Robin Pearce if you have any questions about it.

To ensure our motor controllers are always below their rated temperature of 70C (See data sheet to confirm) we must design a heat sink to keep the motor controllers at least 10C cooler. To do this we will integrate a cold plate into the skin of the aerobody. Since the vehicle will also be racing in very warm climates it is not unreasonable to experience ambient temperatures of 45C outside the car. We must therefore design a heat-sink to function with a 15C temperature delta (Δt = 60C-45C). This page details the required heat flow and thermal conductivity to ensure the motor controllers are kept at a functional temperature. As well the maximum power we can dissipate before our controllers will overheat. This will in turn inform our strategy team what the maximum speed we can travel a hill will be.

Motor Controller Efficiency

The Power loss of the Wave-sculptor 22 motor controller is given by the following equation where:

P_loss = R_eq*I_o² + (α*I_o+β)*V_bus + Cf_eq*V²_bus

Using an approximated value of Pmax for the entire car as 9kW, including mechanical and non-controller electrical losses, the per motor controller peak power is 4.5kW. Meaning the approximated output current will vary between 29.8A - 45A  (Nominal 34.7A) 

From this we can plot the efficiency of the motor controllers versus the voltage and the current using the attached graph. https://www.desmos.com/calculator/kwklxrs8g3

From this plot we can clearly see as power increases the optimal voltage also increases. This optimal voltage range lies in our battery bus range and as such should be relatively optimized to minimize motor controller power losses.

Using the above equation with a peak power of 4500W and a voltage of 80V (lowest reasonable bus voltage) we find the maximum power loss to be 50W. We will assume all of this energy is transferred into heat and as such this will be our required heat flow.

50W waste heat will be used for initial calculations but as is documented below it is not possible to dissipate this much heat through the system so we will plot maximum power to the motor as a function of the air speed assuming all heat must be dissipated through forced air convection through the cold plate.

As such P_loss becomes Q and rearranging using Wolfram we can find Total motor Power, P. For simplicity we will also call V  = E so that speed can remain V. 

P = -(sqrt(V^2 (E (-4 C R V - 4 β R + α^2 V) + 4 Q R)) + α V^2)/(2 R)

From here we can substitute Q with the equation detailed below, where Q is a function of speed  

P = -(sqrt(E^2 (E* (-4* C * R * E - 4 * β * R + α^2 * E) + 4*( 0.037*(V*0.117/0.00001941)^0.8 *0.7241^0.33 *0.0269*0.029*(t)/(0.117) )*R)) + α * E^2)/(2*R)

P = -(sqrt(E^2 (E* (-4* 0.00015625 * 0.0108 * E - 4 * 0.018153 * 0.0108 + 0.003345^2 * E) + 4*( 0.037*(V*0.117/0.00001941)^0.8 *0.7241^0.33 *0.0269*0.029*(t)/(0.117) )*0.0108 )) + 0.003345 * E^2) / (2*0.0108 )

This behemoth of an equation is plotted on the following desmos graph, https://www.desmos.com/calculator/dmdperxjii.

The purple line shows the equation with the 5W of natural convection cooling subtracted from the total power loss.

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.

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.

Heat Conduction

For the purposes of initial design we will assume the cold plate is in direct contact with the outside air. This will most likely not be required and instead it may be in contact with carbon that will then contact outside air. However to show the validation process I have worked through the heat conduction below.

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

.

Since A, Q, and Δt are known we can find the minimum value of k/T. 

k/T = Q/(A*Δt) = 50/(0.029*15) = 115 W/K

Using an approximated value of the aluminum and interface stack up of 200W/m-K we can then find the thickness.

T = k/115 = 200/115 = 1.74m

This value is comically high and as such we do not have to worry if the stack up is exclusively aluminum. However if we are to design the stack with additional materials this math should be revisited.

Heat Convection

For free (non-forced) convection of air we can assume a value of h around 6 W/m^2-K (60C surface temp, 45C ambient temp). From Newtons law of cooling we can calculate the rate of free convection cooling into the car (non-cold-plate side) using the full surface area of the motor controller less the cold plate side (0.029 + 0.024 = 0.053 m^2). We can now calculate natural convection to the ambient air as:

Q_conv = h*A*(Δt)  = 6*0.053*15 = 4.77W

This amount of power dissipation is relatively small and for the purposes of the next analysis we will ignore it.

THINGS GET REAL DICEY FROM HERE ON OUT AND IF I HAVE NOT REMOVED THIS COMMENT I WOULD HIGHLY RECOMMEND READING THE FOLLOWING WITH A GRAIN OF SALT. WHAT IS STATED IS NOT WRONG ITS JUST VERY ODD AND LAID OUT IN A VERY IN-CONCISE MANNER - BE WARY MY FRIENDS. 

*To find if all waste power can be dissipated through forced convection we will calculate the required speed to dissipate 50W.

Rate of convection heat transfer is governed by Newtons law of cooling, 

Q_conv = h*A*(Δt) 

From this equation we will assume all waste heat from the motor controller will be lost through convection and so we can set Q_conv = 50W. The area of the cold plate is taken from the dimensions of the motor controller and is 0.02925 m^2. We can now easily find the required value of the heat transfer coefficient, h.

h = Q_conv/(A*(Δt) = 50/(15*0.02925) = 113.96 W/m^2-K

To find the required value of V we must first understand the heat transfer coefficient, h. The following are required values to find h.

Reynolds number (Re): ratio of inertia forces to viscous forces in the fluid

Re = V*δ/v

Prandtl Number (Pr): a non-dimensional measure of relative thickness of the velocity and thermal boundary layer.

Pr = μ*C_p/k 

Nusselt number (Nu): a non‐dimensional heat transfer coefficient representing the enhancement of heat transfer through a fluid as a result of
convection relative to conduction across the same fluid layer

Nu = h*δ/k 

For a flate plate this becomes → Nu = C*Re^m *Prⁿ where C, m, and n are constants.

Since the flow will be turbulent at this point C = 0.037, m = 0.8, and n = 1/3

A description of the above variables is found in the following table.

*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

From here it is trivial to find V:

Nu = h*L/k = C*Re^m *Prⁿ 

→ Re = (Nu/(C*Prⁿ))^1/m = (h*(L/k)/(C*Prⁿ))^1/m =  (h*(L/k)/(0.037*Pr^0.33))^5/4 

V = Re*v/L = (h*(L/k)/(0.037*Pr^0.33))^5/4 * v/L

V = (114 * (0.117/0.0269) / (0.037*0.7241^0.33))^5/4 * 1.75*10^-5/0.117

V = 26.64 m/s = ~96km/h

This value is extremely fast and as such it is not feasible for us to assume all heat will be dissipated through forced air convection. To find the required speed as a function of Δt I have plotted the following equation in Desmos. https://www.desmos.com/calculator/zc6ihj51gq

V = (Q_conv/(0.029*(Δt) * (0.117/0.0269) / (0.037*0.7241^0.33))^(5/4) * 1.75*10^(-5)/0.117

Since this is not really much use to us we will now rearrange to find Q as a function of speed. From here we can compute what leftover power must be transferred through natural convection or potentially conduction into the carbon fibre aerobody at various speeds (carbon along the plane has relatively great thermal conductivity). The attached desmos graph shows Q as a function of speed https://www.desmos.com/calculator/sahhhpdmej

Q= 0.037*Re^0.8 *Pr^0.33 *k * A*(Δt) / L = 0.037*(V*L/v)^0.8 *Pr^0.33 *k*A*(Δt) / L

Q =  0.037*(V*0.117/0.00001941)^0.8 *0.7241^0.33 *0.0269*0.029*(t)/(0.117) 

For more detailed information on the motor controller see the linked data sheet: