In a manufacturing process, a fluid is transported through a cellar maintained at a temperature of 300 K. The fluid is contained in a pipe having an external diameter of 0.4 m and whose surface has an emissivity of 0.5. To reduce heat losses, the pipe is surrounded by a thin shielding pipe having an ID of 0.5 m and an emissivity of 0.3. The space between the two pipes is effectively evacuated to minimize heat losses and the inside pipe is at a temperature of 550 K. (a) Estimate the heat loss from the liquid per meter length, (b) If the fluid inside the pipe is an oil flowing at a velocity of 1 m/s, calculate the length of pipe for a temperature drop of 1 K.

Question

GIVEN

- Fluid in concentric pipes, with the space between the pipes evacuated, running through a cellar space

- Cellar temperature (T?) = 300 K

- External diameter of inner pipe (D1) = 0.4 m

- Emissivity of outer pipe surface (?1) = 0.5

- Inside diameter of outer pipe (D2) = 0.5 m

- Emissivity of inner pipe (?2) = 0.3

- Inside pipe temperature (T1) = 550 K

FIND

(a) The heat loss from the liquid per meter length (q/L)

(b) The length of pipe for a temperature drop of 1 K if the fluid is oil flowing at a velocity (V) =1 m/s

ASSUMPTIONS

- Steady state

- Convection between the pipes is negligible

- The thermal resistance of the pipe walls is negligible

- The thickness of the outer pipe wall is negligible (Inside surface area = Outside surface area)

- Area of the cellar is large compared to the pipe so that cellar behaves as a blackbody enclosure at T?

- Oil has the thermal properties of unused engine oil

- The temperature of the inner pipe is constant

SKETCH



PROPERTIES AND CONSTANTS

the Stephan-Boltzmann constant



for unused engine oil at 550 K

Density (p) = 742 kg/m3

Specific heat (c) 2998 J/(kg K)

Answer 1

The rate of heat transfer between the pipes is



where F12 is given for infinite concentric cylinders



The rate of heat transfer from the outer pipe to the surroundings is the sum of the rates of convective and radiative heat transfer



An energy balance on the outer pipe yields



Checking the units, then eliminating them for clarity



Since the value of the natural convection heat transfer coefficient, hc, depends on T2, an iterative solution must be used. For the first iteration, let T2 = 400 K.

The Grashof number is



The Nusselt number for this geometry is



Substituting this value into the energy balance yields



By trial and error



(a) The rate of heat transfer from the liquid is



(b) The length of pipe for a temperature drop (?T) of 1 K can be determined from the following



Solving for L