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# Thermal phys: conduction and

1. A cast iron circular water pipe buried in the ground is maintained at 25oC on its inner surface and 15oC on its outer surface. The pipe has an inner diameter of 16cm and an outer diameter of 20cm. Assume one-dimensional steady-state conduction and a thermal conductivity of 10W/mK for the pipe wall material, determine:

a) The conduction heat transfer through an unit length of the pipe wall; and

b) The equation describing the temperature distribution through the pipe wall.

2. A thermocouple is used to measure the temperature of an air-flow in a long large duct having a wall temperature of 500K. The temperature of the air flow is 600K. Estimate the temperature reading indicated by the thermocouple if the convection heat transfer coefficient between the thermocouple and the airflow is 20W/m^2K. Assume the thermocouple to be a black body.

ANS: 266oC

Thank you very much!

### 1 Answer

- 天同Lv 77 years agoFavorite Answer
1. The heat conduction equation in one dimension is,

Q = kA.(dT/dr)

where Q is the rate of flow of heat by conduction

k is the thermal conductivity of the material

A is the cross-sectional area

dT/dr is the temperature gradient.

For circular pipe, we have,

Q = k[(2.pi.r).L].(dT/dr) where L is the length of the pipe and r is the radial distance from the centre of the pipe.

Hence, Q(dr/r) = k.(2.pi.L)(dT)

Integrate both sides, with r from ri, the inner radius of the pipe, to an arbitrary radial distance r; and T from Ti (temperature of inner surface of pipe) to T (temperature at distance r).

Q[ln(r/ro)] = k(2.pi.L)(T - Ti)

i.e. Q = k(2.pi.L).(T - Ti)/ln(r/ri)

For unit length of pipe, L = 1 m

Q = 10 x (2.pi) x (25 - 15)/ln(20/16) w = 2816 w

2. Is there a diagram for this problem? Where is the thermocouple placed?