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Chapter 4: Problem 76

Obtain expressions for the rate of change in mass of the control volume shown, as well as the horizontal and vertical forces required to hold it in place, in terms of \(p_{1}, A_{1}, V_{1}, p_{2}\) \(A_{2}, V_{2}, p_{3}, A_{3}, V_{3}, p_{4}, A_{4}, V_{4},\) and the constant density \(\rho\).

Short Answer

Expert verified
The expressions for the rate of mass change and horizontal and vertical forces in terms of the given variables are, respectively: \(\frac{dm}{dt} = \dot{m}_1 - \dot{m}_2 - \dot{m}_3 + \dot{m}_4\), the horizontal force \(F_x = \dot{m}_1*V_{1x} - \dot{m}_2*V_{2x} - \dot{m}_3*V_{3x} + \dot{m}_4*V_{4x}\) and vertical force \(F_y = \dot{m}_1*V_{1y} - \dot{m}_2*V_{2y} - \dot{m}_3*V_{3y} + \dot{m}_4*V_{4y}\), where the dots represent the time derivative.

Step by step solution

01

Calculate the rate of mass change

We know that the rate of mass change is flow rate in minus flow rate out. We need to find the mass flow rates at each point by using the equation \(\dot{m} = \rho A V\). The total rate of mass change will be \(\frac{dm}{dt} = \dot{m}_1 - \dot{m}_2 - \dot{m}_3 + \dot{m}_4\). Substitute each mass flow rate in the equation to determine the rate of mass change, where each \(\dot{m}_i\) represents the mass flow rate at each point \(i\).
02

Calculate horizontal and vertical forces

The horizontal force is the change in momentum in the x-direction. Use \(F_x = \frac{d(mV_x)}{dt}\). Use the mass flow rate and velocity at each point to find the contribution to the force. The total force will be the sum of the forces at each point. The vertical force is calculated the same way as the horizontal force, but using the y-component of the velocity, i.e. \(F_y = \frac{d(mV_y)}{dt}\). This is because the force required to hold the control volume in place is in both the x and y direction.
03

Express the forces

Combine forces from all points to express the forces in terms of the question variables. \(F_x = \dot{m}_1*V_{1x} - \dot{m}_2*V_{2x} - \dot{m}_3*V_{3x} + \dot{m}_4*V_{4x}\) and \(F_y = \dot{m}_1*V_{1y} - \dot{m}_2*V_{2y} - \dot{m}_3*V_{3y} + \dot{m}_4*V_{4y}\). This method considers the conservation of momentum, which includes contributions from both inertial and pressure forces, to hold the control volume in place.

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