Fluid Dynamics Problem

Question

A vertical circular stack 100 ft high converges uniformly from a diameter of 20 ft at the bottom to 16 ft at the top. Coal gas with a unit weight of 0.030 pcf enters the bottom of the stack with a velocity of 10 fps. The unit weight of the gas increases uniformly to 0.042 pcf at the top. What is the mean velocity 25 ft above the bottom of the stack?

Answer 1

Converting velocity in to volumetric flow rate

At the bottom of stack:$$v_1 = \frac{Q_1}{A_1}$$

From this find $Q_1$

Thus $$Q_1 = 10\times \pi\times 10^2 = 1000\pi$$

Now since the unit weight changes, equate the mass flow rate such as below:

$$\rho_1\times Q_1 = \rho_2\times Q_2$$

Unit weight changes uniformly down the stack at the rate of $\frac{(.042-.030)}{100} = 0.00012$

At the stack height of 25 ft , the unit weight $= .03+0.00012*25 = 0.033$

Thus $\rho_2 = 0.033$. Apply this in the equation above to get $Q_2$

$$Q_2 = \frac{0.03*1000\pi}{0.033} = 909.09\pi$$

Now $$V_2 = \frac{Q_2}{A_2} = \frac{909.09\pi}{9.5^2\pi}$$ If you solve this you get $v_2 = 10.07$ fps