Consider a large 5cm thick brass plate (k=111W/m C) in whichheat is generated un
ID: 1815511 • Letter: C
Question
Consider a large 5cm thick brass plate (k=111W/m C) in whichheat is generated uniformly at a rate of(2x105W/m3). One side of the plate isinsulated while the other side is exposed to an environment at 25Cwith a heat transfer coefficient of 44W/m2C. Explainwhere in the plate the highest and lowest temperatures will occur,and determine their values.Explanation / Answer
Consider a large 5cm thick brass plate (k=111W/m C) in which heatis generated uniformly at a rate of(2x105W/m3). One side of the plate isinsulated while the other side is exposed to an environment at 25Cwith a heat transfer coefficient of 44W/m2C. Explainwhere in the plate the highest and lowest temperatures will occur,and determine their values. *cp*dT/dt = k*d2T/dx2 + generation =0 b/c dT/dt = 0 (steady state) k*d2T/dx2 + generation =0 generation = 2e5 W/m3 and boundary conditions dT/dx = 0 (insulated) at x =0 -k*dT/dx = h*(T -T) at x = 0.05 Solve k*d2T/dx2 + gen = 0 d2T/dx2 = -gen/k = C1 dT/dx = C1*x + C2 T = (C1/2)*x2 + C2x + C3 C1 = (-gen/k) = (-2e5 / 111) = -1801.8018 Use dT/dx = 0 at x = 0 dT/dx = C1*x + C2 => C1*0 + C2 = 0 => C2 = 0 Now T looks like this T = (C1/2)*x2 +C3 => dT/dx = C1*x -k*dT/dx = h*(T -T) at x = 0.05 -k*C1*x = h*((C1/2)*x2 + C3 - T)= gen*x recall C1 = -gen/k Consider a large 5cm thick brass plate (k=111W/m C) in which heatis generated uniformly at a rate of(2x105W/m3). One side of the plate isinsulated while the other side is exposed to an environment at 25Cwith a heat transfer coefficient of 44W/m2C. Explainwhere in the plate the highest and lowest temperatures will occur,and determine their values. (44)*(-1801.8018/2 *0.05^2 + C3 - 25) = 2e5 *(0.05) C3 = 254.52498 Thus T = (C1/2)*x2 + C2x + C3 =(-1801.8018/2)*x2 + 0*x + 254.52498 T = -900.9009*x2 + 254.52498 Largest T is at x = 0 (T = 254.5 C Smallest T is at x = 0.05, T = 252.3 C Hope that helps
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