- #1
weirdoguy
- 1,134
- 1,086
- Homework Statement
- Factory chimney has a height of ##h=22,4m##. Its side inlet opening is tightly closed with a cover whose area is ##S=1m^2##. Atmospheric air temperature is ##t_0=0^\circ C## and its pressure is ##P_0=10^5 Pa##. Calculate the value of the average air temperature in the chimney, if it is known that due to the temperature difference, the force ##F=85N## acts on the cover.
- Relevant Equations
- Hydrostatic pressure; relation between density of air in different temperatures
Kind of similar question was once on polish high school ("matura") exam, and the solution that the authors gave was:
Pressure at the bottom of the chimney inside of it is ##p_{top}+\rho_1gh##, and outside ##p_{top}+\rho_2gh## where:
##p_{top}## is the atmospheric pressure at the height of ##h##,
##\rho_1## is the density of air inside the chimney,
##\rho_2## is the density of air outside.
Force is then ##F=|\rho_1-\rho_2|ghS##. But in that situation, densities were given in the problem. Here they are not. I guess we can use the equation:
##\frac{\rho_1}{\rho_2}=\frac{T_2}{T_1}##, where density in temperature of ##0^\circ C## can be found in the tables, or somewhere (it's somewhere around ##1,2kg/m^3## if I remember correctly). But how to use this ##P_0## given in the problem? At what level is it measured? In yet another exercise it was said that ##P_0## is measured at the bottom, but still using it forces me to use mollar mass of air through Clapeyron equation...
In yet another problem book I found similar exercise in which authors use coefficient of thermal expansion...
I really don't know where to go with this.
Pressure at the bottom of the chimney inside of it is ##p_{top}+\rho_1gh##, and outside ##p_{top}+\rho_2gh## where:
##p_{top}## is the atmospheric pressure at the height of ##h##,
##\rho_1## is the density of air inside the chimney,
##\rho_2## is the density of air outside.
Force is then ##F=|\rho_1-\rho_2|ghS##. But in that situation, densities were given in the problem. Here they are not. I guess we can use the equation:
##\frac{\rho_1}{\rho_2}=\frac{T_2}{T_1}##, where density in temperature of ##0^\circ C## can be found in the tables, or somewhere (it's somewhere around ##1,2kg/m^3## if I remember correctly). But how to use this ##P_0## given in the problem? At what level is it measured? In yet another exercise it was said that ##P_0## is measured at the bottom, but still using it forces me to use mollar mass of air through Clapeyron equation...
In yet another problem book I found similar exercise in which authors use coefficient of thermal expansion...
I really don't know where to go with this.
I guess I don't know how to finish this exercise, since I see different options for that. So my question is what would you do? I would go with the density, without using ##P_0##, but autors gave it for a reason. So maybe I should use this molar mass. Unfortunately I don't have the book in which this exercise is, my student got that as a homework. I asked on polish forum if someone know where is it from, I would like to see what answer authors gave.