Question

Homework #4 A motor is constructed using windings that surround laminated iron poles. You have been asked to estimate the maximum temperature that will occur within the windings. The windings and poles are both approximated as being cylindrical, as shown below. 2 cm F-1em L-2 cm T-25°C h-25 W/m-K windings T50°C k-1 W/m-K -1x10 Wim The windings are a complicated composite formed from copper conductor, insulation and air that fills the gaps between adjacent wires. However, in most models, the windings are represented by a solid with equivalent properties that account for this underlying structure. You can therefore consider the windings to be a solid. The electrical current in the windings causes an ohmic dissipation that can be modeled as a uniform volumetric generation rate of g-1x10 W/m. The conductivity of the windings is k-1.0 W/m-K. The inner radius of the windings is rn 1.0 cm and the outer radius is roa -2.0 cm. The windings are L 2.0 cm long and the upper and lower surfaces may be assumed to be insulated so that the temperature in the windings varies only in the radial direction. The stator pole is conductive and cooled externally; therefore, you can assume that the stator tooth has a uniform temperature of Tole 50°C. Neglect any contact resistance between the inner radius of the winding and the pole; therefore, the temperature of the windings at r=rin is Tole The outer radius of the windings is exposed to air at Tair 25°C with a heat transfer coefficient of h-25 W/m2-K. a.) Derive the governing differential equation for the temperature within the windings (i.e., the differential equation that is valid from rin<r<r) b.) Specify the boundary conditions for your differential equation. c.) Solve the governing differential equation that you derived in part (a) by integrating twice. You should end up with a solution that involves two constants of integration, Ci and C2. d.) Substitute your answer from part (c) into the boundary conditions specified in part (b) to obtain two equations for your two unknown constants of integration, Ci and C2. e.) Implement your results from (c) and (d) in EES and prepare a plot of the temperature in the stator as a function of radius.

Answer 1

(a) Assuming differential control volume of windings artdr The energy balance on control volume is given by: g +g g. Heat into the differential volume qtdrHeat out from differential volume T Temperature thermal energy generation within control volume expressed as the product of the control volume and the volumetricrate of generation. Term qrtdr can be expand using Taylor's series, neglecting higherorder terms and taking the limit as dr approachesto zero. dr + dr dar dr..(1) Hence, g dr g' (2TorL)dr..(2) We know Fourier's law for conduction: dT -k (2TrL) (3) = dr Substitute equation (3) and (2) into equation (1): dT dr...4) d -k(2TrL) dr g (2TrL)dr dr. dT d .(5) dr dr.

(b) Boundary condition at the inner radius is: TTin At interface of air and outer radius: qrront T _ 4convection Using Fourier's law and Newton's law of cooling and applying to above equation: dT -k drノャニrout = h(7,rut- Tatr) 6) (c) Rearranging equation (5) and integrating: dr k dr We get g'2 dT 2k dr Temperature gradient is given by: dT (a) + 2k dr Rearranging above equation and integrating dr _ 2k f CInr)C2. (7) T = 4k

(d) Substituting above equation (7)for T into equation (6) g"' Tin CIn(r) 4k C2 Tpole _ Substituting above equation (7) and (a) into equation (6) 2 Tout C1 + out out C In(rout)C2- Tatr 2k 4k (e) Belowfigure shows Temperature Vs Radius plot from EES 95 90 85 80 75 70 65 60 55 50 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 CC LC