# THE hard Chimney Physics problem from the Ipho/JEE

TLDRThis video script delves into the physics of chimney efficiency, using Bernoulli's equation to calculate the minimum height required for effective smoke expulsion. It begins with the volume flow rate and the negligible speed of particles within the furnace, then applies principles of fluid dynamics to derive the exit velocity of smoke. The script further explores the relationship between the chimney's height, the densities of smoke and air, and temperature differences, providing a formula for the minimum height based on these variables. The video concludes with an encouragement to explore more physics problems from the International Physics Olympiad.

###### Takeaways

- π The video discusses the minimum height of a chimney for efficient operation, based on a question from the International Physics Olympiad.
- π The script translates the problem into a mathematical model, starting with the volume flow rate equation \( \frac{dV}{dt} = B \).
- π‘ It uses Bernoulli's principle to calculate the speed of the smoke exiting the chimney, considering the pressure and velocity at two points.
- π The script assumes negligible particle speed within the furnace and focuses on the minimum chimney height required for smoke to escape.
- βοΈ The pressure of the smoke at the chimney's exit must be at least equal to the atmospheric pressure at that height for efficient operation.
- πͺ The equation derived relates the chimney's height to the densities of the smoke and air, and the gravitational force.
- π The final expression for the chimney's minimum height is given in terms of the volume flow rate, area, and the difference in temperature between the smoke and the ambient air.
- π‘οΈ The script assumes the behavior of an ideal gas inside the chimney, allowing the use of the equation \( PV = nRT \) to relate density and temperature.
- π The video concludes by expressing the minimum height in terms of the temperature difference, assuming constant pressure and molar mass.
- π€ The presenter encourages viewers who enjoy physics problems to seek out more challenging questions from the International Physics Olympiad.
- π₯ The video script is presented in an educational and engaging manner, with a focus on the beauty of physics in solving real-world problems.

###### Q & A

### What is the volume flow rate of smoke produced by the chimney per unit time?

-The volume flow rate of smoke produced by the chimney per unit time is given to be B, as stated in the script.

### What principle is used to calculate the speed of the smoke at the top of the chimney?

-Bernoulli's principle is used to calculate the speed of the smoke at which it exits the chimney.

### Why is Bernoulli's equation applied at two points, one at height H and the other slightly below?

-Bernoulli's equation is applied at two points to consider the pressure and velocity differences between the top of the chimney and a point just below it, assuming negligible pressure variation over a small distance.

### What is the significance of the term 'rho G times h' in the Bernoulli's equation context?

-The term 'rho G times h' represents the hydrostatic pressure difference due to the height of the fluid column, where rho is the density, G is the acceleration due to gravity, and h is the height.

### Why is the speed of particles within the furnace considered negligible for this calculation?

-The speed of particles within the furnace is considered negligible because the focus is on the speed of the smoke as it exits the chimney, which is influenced by the pressure differences and not the initial speed within the furnace.

### What is the minimum condition for the pressure of the smoke at the top of the chimney?

-The minimum condition for the pressure of the smoke at the top of the chimney is that it must be at least equal to or greater than the atmospheric pressure at that height to ensure efficient operation.

### How is the final expression for the velocity of the smoke (Vh) derived?

-The final expression for the velocity of the smoke (Vh) is derived by simplifying the Bernoulli's equation and considering the pressure differences, resulting in Vh = sqrt(2gH * (rho_air/rho_smoke - 1)), where g is the acceleration due to gravity.

### What is the relationship between the volume flow rate (B) and the velocity of the smoke (V)?

-The relationship between the volume flow rate (B) and the velocity of the smoke (V) is given by B = A * V, where A is the cross-sectional area of the chimney.

### How does the temperature difference between the ambient air and the smoke affect the minimum height of the chimney?

-The temperature difference affects the minimum height of the chimney by influencing the density of the smoke and air, which in turn affects the pressure difference and the required height for the smoke to escape efficiently.

### What assumptions are made when deriving the expression for the minimum height of the chimney in terms of temperature differences?

-The assumptions made include the behavior of the gases as ideal gases, constant pressure, and that the molar mass and the gas constant are constants, leading to the relationship rho_air * T_air = rho_smoke * T_smoke.

###### Outlines

##### π Applying Physics to Determine Minimum Chimney Height

This paragraph delves into the physics behind the efficiency of a chimney's operation, referencing a problem from the International Physics Olympiad and a variation from JEE Advance 2023. It introduces the concept of using Bernoulli's equation to calculate the speed at which smoke exits the chimney. The discussion involves setting up the equation at two points, one at the top of the chimney and another slightly below, to account for pressure differences and the height of the chimney. The key takeaway is the importance of the smoke's pressure being at least equal to the surrounding air pressure for efficient operation, leading to the derivation of a formula for the minimum chimney height based on the density of the smoke and air.

##### π Calculating Smoke Velocity and Volume Flow Rate

The second paragraph focuses on deriving the velocity of smoke as it exits the chimney and relating it to the volume flow rate. It starts by simplifying Bernoulli's equation to find the velocity (Vh) and then connects this to the volume flow rate (B), which is given as a constant. The summary explains how the area of the chimney (a) and the velocity (V) are related to the volume flow rate, leading to an expression for the chimney's height in terms of the densities of air and smoke. The paragraph also introduces the concept of temperature differences and how they relate to the densities, ultimately providing an algebraic manipulation to express the minimum chimney height in terms of temperature change rather than density differences.

##### π Final Expression for Minimum Chimney Height

In the concluding paragraph, the presenter wraps up the derivation by providing the final expression for the minimum height of the chimney, which is expressed in terms of the temperature difference between the ambient air and the smoke. The summary emphasizes the assumptions made, such as constant pressure and ideal gas behavior, and how these lead to the relationship between the densities of air and smoke being proportional to their respective temperatures. The final formula for the minimum height (H) is presented, highlighting the importance of understanding temperature gradients for chimney design.

###### Mindmap

###### Keywords

##### π‘Chimney Height

##### π‘Bernoulli's Equation

##### π‘Volume Flow Rate

##### π‘Pressure Variation

##### π‘Density

##### π‘Ideal Gas Law

##### π‘Hydrostatic Equilibrium

##### π‘Speed of Smoke

##### π‘Temperature Gradient

##### π‘Molar Mass

###### Highlights

The problem involves determining the minimum height of a chimney for it to operate efficiently.

The volume flow rate of gas or smoke produced by the chimney is represented as \( \frac{dV}{dt} = B \).

The speed of particles within the furnace is negligible, simplifying the calculations.

Bernoulli's equation is applied at two points: one at height \( H \) above a reference point, and another just below it.

The pressure of the smoke exiting the chimney must be at least equal to the pressure of the surrounding air at the same height.

The equation is rearranged to express the velocity \( v_H \) of the smoke at the top of the chimney: \( v_H = \sqrt{\frac{2gH(\rho_{\text{air}} - \rho_{\text{smoke}})}{\rho_{\text{smoke}}}} \).

The minimum height is derived using the ideal gas law, linking the density ratio to the temperature difference between the smoke and ambient air.

The final expression for the minimum height \( H \) is \( H = \frac{B^2 T_{\text{air}}}{A^2 \cdot 2g(T_{\text{smoke}} - T_{\text{air}})} \).

The approach assumes constant pressure and uses the relationship \( \rho_{\text{air}} \cdot T_{\text{air}} = \rho_{\text{smoke}} \cdot T_{\text{smoke}} \).

The analysis links chimney height with the temperature gradient, which is critical for efficient chimney design.

This problem is derived from the International Physics Olympiad, showing its complexity and educational value.

A variation of this problem appeared in the JEE Advance 2023 exam, emphasizing its relevance in competitive exams.

The transcript demonstrates the process of converting a real-world problem into a mathematical model.

Careful assumptions are made to simplify the problem, such as negligible speed within the furnace and infinitesimal height differences.

The solution combines principles from fluid dynamics, thermodynamics, and mathematical analysis.

###### Transcripts

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