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When modeling fluid flows with small changes in
Chapter 2, Problem 46P(choose chapter or problem)
Problem 46P
When modeling fluid flows with small changes in temperature and pressure, the Boussinesq approximation is often used in which the fluid density is assumed to vary linearly with changes in temperature. The Boussinesq approximation is ρ = ρ()[1 –β(T– To)], where β is assumed to be constant over the given temperature range; β is evaluated at reference temperature T0, taken as some average or mid-value temperature in the flow; and ρ0 is a reference density, also evaluated at T0. The Boussinesq approximation is used to model a flow of air at nearly constant pressure,P = 95.0 kPa, but the temperature varies between 20°C and 60°C. Using the mid-way point (40°C) as the reference temperature, calculate the density at the two temperature extremes using the Boussinesq approximation, and compare with the actual density at these two temperatures obtained from the ideal gas law. In particular, for both temperatures calculate the percentage error caused by the Boussinesq approximation.
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QUESTION:
Problem 46P
When modeling fluid flows with small changes in temperature and pressure, the Boussinesq approximation is often used in which the fluid density is assumed to vary linearly with changes in temperature. The Boussinesq approximation is ρ = ρ()[1 –β(T– To)], where β is assumed to be constant over the given temperature range; β is evaluated at reference temperature T0, taken as some average or mid-value temperature in the flow; and ρ0 is a reference density, also evaluated at T0. The Boussinesq approximation is used to model a flow of air at nearly constant pressure,P = 95.0 kPa, but the temperature varies between 20°C and 60°C. Using the mid-way point (40°C) as the reference temperature, calculate the density at the two temperature extremes using the Boussinesq approximation, and compare with the actual density at these two temperatures obtained from the ideal gas law. In particular, for both temperatures calculate the percentage error caused by the Boussinesq approximation.
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