Thermal management of batteries is required to prolong the battery lifetime, increase the energy storage capacity,...

  

 

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Thermal management of batteries is required to prolong the battery lifetime, increase the energy storage capacity, and prevent catastrophic failure due to overheating. A battery of thickness L = 2 cm and thermal conductivity k = 1.6 W/mK has a position-independent volumetric energy generation rate g = 4 * 105 W/m³. At x = 0 the battery is connected to a wall and kept at a temperature T = 300 K. The surface of the battery at x = L is exposed to a fluid with a convection coefficient h = 120 W/m²K and fluid temperature To = 300 K. This is a steady-state problem, and heat flow is one-dimensional in the x direction. a) Solve the governing heat conduction differential equation to find a general expression for T(x). Your expression for T(x) should depend only on x, g, k, and two constants of integration (C₁, C₂). b) Apply the boundary conditions to solve for C₁ and C₂. Give a final analytical expression for the temperature profile T(x). Your answer may depend only on x, g, k, L, T, and h. c) Calculate the heat flux in the battery at x = 0, q(x = 0). Give a numerical answer (units: W/m²). (Note: Your answer for q should be a negative number, because heat is flowing in the negative x direction out of the battery and into the wall). d) Using Matlab or a similar software, plot T as a function of x from x = 0 to x = L. Make sure to label the axes of your graph. Comment on the main takeaways from your plot. Too battery 9, k X L Ch, Too Thermal management of batteries is required to prolong the battery lifetime, increase the energy storage capacity, and prevent catastrophic failure due to overheating. A battery of thickness L = 2 cm and thermal conductivity k = 1.6 W/mK has a position-independent volumetric energy generation rate g = 4 * 105 W/m³. At x = 0 the battery is connected to a wall and kept at a temperature T = 300 K. The surface of the battery at x = L is exposed to a fluid with a convection coefficient h = 120 W/m²K and fluid temperature To = 300 K. This is a steady-state problem, and heat flow is one-dimensional in the x direction. a) Solve the governing heat conduction differential equation to find a general expression for T(x). Your expression for T(x) should depend only on x, g, k, and two constants of integration (C₁, C₂). b) Apply the boundary conditions to solve for C₁ and C₂. Give a final analytical expression for the temperature profile T(x). Your answer may depend only on x, g, k, L, T, and h. c) Calculate the heat flux in the battery at x = 0, q(x = 0). Give a numerical answer (units: W/m²). (Note: Your answer for q should be a negative number, because heat is flowing in the negative x direction out of the battery and into the wall). d) Using Matlab or a similar software, plot T as a function of x from x = 0 to x = L. Make sure to label the axes of your graph. Comment on the main takeaways from your plot. Too battery 9, k X L Ch, Too