Exhaust gases having properties similar to dry air enter a thin-walled cylindrical exhaust stack at 800 K. The stack is made of steel and is 8-m-tall and 0.5 m inside diameter. If the gas flow rate is 0.5 kg/s and the heat transfer coefficient at the outer surface is 16 W/(m2 K), estimate the outlet temperature of the exhaust gas if the ambient temperature is 280 K.

Question

GIVEN

• Gas properties are similar to dry air

• Gas entrance temperature (Tb,in) = 800 K

• Length of stack (L) = 8 m

• Diameter of stack (D) = 0.5 m

• Mass flow rate (m

) = 0.5 kg/s

• Heat transfer coefficient on the outer surface ( h ,c o ) = 16 W/(m2 K)

• Ambient temperature (T?) 280 K FIND

• The outlet temperature of the exhaust gas (Tb,out)

ASSUMPTIONS

• Radiation heat transfer is negligible

• Natural convection can be neglected

• The inlet to the stack is sharp-edged

• Thermal resistance of the stack wall is negligible

SKETCH

Answer 1

For this problem, neither the heat flux nor the surface temperature will be constant. However, the ambient temperature will be constant, therefore, can be applied by replacing the surface temperature (Ts) with the constant ambient temperature (T?) and replacing hc with U where

U = Overall heat transfer coefficient = 1



This results in the following version



The internal heat transfer coefficient and the average fluid properties will depend on the outlet bulk fluid temperature, therefore, an iterative procedure is required. For the first iteration, let Tb,out = 500 K.

From for dry air at the average bulk temperature of 650 K

Specific Heat (cp) = 1056 J/(kg K)

Thermal conductivity (k) = 0.0472 W/(m K)

Absolute viscosity (?) = 31.965 × 10–6 (Ns)/m2

Prandtl number (Pr) = 0.71

The Reynolds number for flow in the stack is



L/D = (8 m)/(0.5) = 16

Since 2 < L/D < 20, the flow will not be fully developed, therefore, the correlation of Molki and

Sparrow for sharp-edged inlets, will be used to correct the correlation of Dittus-Boelter,



Another iteration using the same procedure yields

Average bulk temperature = 762 K

Specific Heat (cp) = 1074 J/(kg K)

Thermal conductivity (k) = 0.0534 W/(m K)

Absolute viscosity (?) = 35.460 × 10–6 (Ns)/m2

Prandtl number (Pr) = 0.72

Reynolds number (ReD) = 3.59 × 104

Heat transfer coefficient ( ci h ) = 12.1 W/(m2 K)

Outlet temperature (Tb,out) = 722 K

The outlet gas temperature = 722 K