Engenharia

PROBLEM 2.1
KNOWN: Steady-state, one-dimensional heat conduction through an axisymmetric shape. FIND: Sketch temperature distribution and explain shape of curve. SCHEMATIC:

ASSUMPTIONS: (1) Steady-state, one-dimensional conduction, (2) Constant properties, (3) No internal heat generation.

  ANALYSIS: Performing an energy balance on the object according to Eq. 1.11a, E in − E out = 0, it follows that   E in − E out = q x and that q x ≠ q x x . That is, the heat rate within the object is everywhere constant. From Fourier’s law,

$

q x = − kA x

dT , dx

and since qx and k are both constants, it follows that

Ax

dT = Constant. dx

That is, the product of the cross-sectional area normal to the heat rate and temperature gradient remains a constant and independent of distance x. It follows that since Ax increases with x, then dT/dx must decrease with increasing x. Hence, the temperature distribution appears as shown above. COMMENTS: (1) Be sure to recognize that dT/dx is the slope of the temperature distribution. (2) What would the distribution be when T2 > T1? (3) How does the heat flux, q ′′ , vary with distance? x

PROBLEM 2.2
KNOWN: Hot water pipe covered with thick layer of insulation. FIND: Sketch temperature distribution and give brief explanation to justify shape. SCHEMATIC:

ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional (radial) conduction, (3) No internal heat generation, (4) Insulation has uniform properties independent of temperature and position. ANALYSIS: Fourier’s law, Eq. 2.1, for this one-dimensional (cylindrical) radial system has the form

q r = − kA r

dT dT = − k 2πr dr dr

1 6

where A r = 2πr and  is the axial length of the pipe-insulation system. Recognize that for steadystate conditions with no internal heat generation, an energy balance on the system requires     E in = E out since E g = E st = 0. Hence

qr = Constant.
That is, qr is independent of radius (r). Since the thermal