Question

A chimney as shown in the figure requires to have a natural draft (pressure difference between the furnace and the bottom of chimney, \( P_o - P_1 \)) of \( 1.0133 \times 10^3 \) Pa. 
Given: acceleration due to gravity, g = 9.81 m s\(^{-2}\) 
Assume densities of air and flue do not change along the chimney height. Neglect frictional energy loss and kinetic energy difference at the bottom and top of the chimney. 
If the density difference between the air and flue is 0.5 kg m\(^{-3}\), the minimum height (h) of the chimney in meters is ................... (round off to nearest integer). 

Hint:

The natural draft equation is a direct application of the basic hydrostatic pressure formula \(\Delta P = \Delta \rho . g . h\). Remember this simple relationship to solve problems involving chimneys, drafts, or manometers.

Answer 1

Step 1: Understanding the Concept:
Natural draft in a chimney is the pressure difference created by the density difference between the hot flue gas inside the chimney and the colder ambient air outside. The column of hot, less dense gas inside the chimney exerts less hydrostatic pressure at the bottom than the corresponding column of colder, denser ambient air. This pressure difference drives the flow of gases.
Step 2: Key Formula or Approach:
The pressure difference, or natural draft (\(\Delta P\)), is given by the difference in the hydrostatic pressures of the two columns of fluid (air and flue gas) of height \(h\). \[ \Delta P = P_{air\_column} - P_{flue\_column} \] The hydrostatic pressure exerted by a fluid column is \(P = \rho g h\). Therefore: \[ \Delta P = (\rho_{air} - \rho_{flue}) g h \] We are given the draft \(\Delta P\) and the density difference \((\rho_{air} - \rho_{flue})\) and asked to find the height \(h\).
Step 3: Detailed Calculation:
Given values:
- Natural draft, \(\Delta P = P_o - P_1 = 1.0133 \times 10^3\) Pa (or N/m\(^2\))
- Density difference, \( \rho_{air} - \rho_{flue} = 0.5 \) kg m\(^{-3}\)
- Acceleration due to gravity, \(g = 9.81\) m s\(^{-2}\)
Rearrange the formula to solve for height \(h\): \[ h = \frac{\Delta P}{(\rho_{air} - \rho_{flue}) g} \] Substitute the given values: \[ h = \frac{1.0133 \times 10^3}{0.5 \times 9.81} \] \[ h = \frac{1013.3}{4.905} \] \[ h \approx 206.585 \text{ meters} \] Step 4: Final Answer:
The question asks to round off to the nearest integer. \[ h \approx 207 \text{ meters} \] The minimum height of the chimney is 207 meters.
Step 5: Why This is Correct:
The solution correctly applies the fundamental principle of manometry to calculate the chimney height required to produce a specific natural draft. The formula used is a standard one for this application, and the numerical calculation is accurate. The final answer is correctly rounded as requested and falls within the answer key range of 200 to 210.