An electrical cable, experiencing uniform volumetric generation q ˙ , is half buried in an insulating material while the upper surface is exposed to a convection process T ∞ , h . (a) Derive the explicit, finite-difference equations for an interior node ( m , n ), the center node m = 0 , and the outer surface nodes ( M , n ) for the convection and insulated boundaries. (b) Obtain the stability criterion for each of the finite-difference equations. Identify the most restrictive criterion.
An electrical cable, experiencing uniform volumetric generation q ˙ , is half buried in an insulating material while the upper surface is exposed to a convection process T ∞ , h . (a) Derive the explicit, finite-difference equations for an interior node ( m , n ), the center node m = 0 , and the outer surface nodes ( M , n ) for the convection and insulated boundaries. (b) Obtain the stability criterion for each of the finite-difference equations. Identify the most restrictive criterion.
Solution Summary: The author compares two rods of the same diameter but different materials used.
An electrical cable, experiencing uniform volumetric generation
q
˙
, is half buried in an insulating material while the upper surface is exposed to a convection process
T
∞
,
h
.
(a) Derive the explicit, finite-difference equations for an interior node (m, n ), the center node
m
=
0
, and the outer surface nodes (M, n ) for the convection and insulated boundaries.
(b) Obtain the stability criterion for each of the finite-difference equations. Identify the most restrictive criterion.
A long wire of diameter D = 2 mm is submerged in an oil bath of temperature T∞ = 23°C. The wire has an electrical resistance per unit length of Re′=0.01 Ω/m. If a current of I = 180 A flows through the wire and the convection coefficient is h = 529 W/m2 · K, what is the steady-state temperature of the wire? From the time the current is applied, how long does it take for the wire to reach a temperature that is within 2°C of the steady-state value? The properties of the wire are ρ = 2,334 kg/m3, c = 537 J/kg · K, and k = 43 W/m · K.
Presents the diagram of the problem, necessary formulas, clearance and numerical solution:
Two heat reservoirs with respective temperatures of 325 and 275 K are brought into contact by an iron rod 200 cm long and 24 cm2 in cross section. Calculate the heat flux between the reservoirs when the system reaches its steady state. The thermal conductivity of iron at 25 ◦C is 79.5 W/m K.
A silicon chip is encapsulated so that, under steady-state conditions, all the power dissipated by it is transferred by convection for a fluid current with h=1000 W/(m2.K) and Tꚙ=25oC. An aluminum plate (k=237 W/(mK)), 2 mm thick, is placed on the chip surface, the contact resistance on the chip/aluminum interface is 0.5x10-4 (m2.K)/W. If the chip area is 100 mm2 and the maximum allowable temperature is 85 oC, what is the maximum allowable power dissipation on the chip?
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