Quiz: Physics-based Modeling

1. Which of these is not a conserved quantity as a basis for a balance equation?

A. Momentum
Incorrect. Momentum is a conserved quantity
B. Species
Incorrect. The moles of a compound (species) is a conserved quantity but can be changed with reaction. A balance equation tracks the change in species.
C. Force
Correct. Force is not a conserved quantity. An object in motion tends to stay in motion and forces acting on that object modify the trajectory. To track the motion of an object, you can start with a momentum balance as the conserved quantity for the balance equation.
D. Energy
Incorrect. Energy is a conserved quantity. One generation term may be nuclear applications that produce energy from mass during fission or fusion.

2. In some large systems such as reservoirs or algae cultivation pools, evaporation plays a significant role. How would its effect best be classified?

A. Controller Output / Manipulated Variable
Incorrect. The evaporation cannot be changed by an actuator. It is a process disturbance.
B. Disturbance
Correct. Changes to evaporation occur from external forces. It is an exogenous (independent) input to the system.
C. Process Variable / Controlled Variable
Incorrect. The process variable (PV) of a control loop or the controlled variable (CV) of a predictive controller have a target set point and are actively regulated by adjusting an actuator. The level in the pool may be a PV / CV but the evaporation is a disturance to regulating that level to a target value.

3. The mass balance for a vertical cylindrical tank assumes a constant cross-sectional area A:

$$\rho A \frac{dh}{dt} = \dot m_{in} - \dot m_{out}$$

with `A=\pi r^2` and radius `r`. The volume of liquid in the vertical cylinder is `V=A h`. What is the form of the differential equation if the cylinder is placed horizontally?

A. $$\rho A \frac{dh}{dt} = \dot m_{in} - \dot m_{out}$$
Incorrect. The area (A) depends on height. This formula is for a vertical cylinder.
B. $$\pi h^2 \frac{dh}{dt} = \dot m_{in} - \dot m_{out}$$
Incorrect. The area changes with height must be integrated from 0 to the new height. See Volume of a Horizontal Cylinder for the derivation.
C. $$\rho \frac{dV}{dt} = \dot m_{in} - \dot m_{out}$$
Correct. It is more complicated to create a differential equation that has `{dh}/{dt}`. Instead, solve for volume with the mass balance equation and then solve for height with the equation:
$$V=L \left(r^2 \cos^{-1}\left(\frac{r-h}{r}\right)-\left(r-h\right)\sqrt{2rh-h^2}\right)$$
where L is the length of the cylinder in the horizontal position. The volume of the water varies with height

4. A simulation of the ice temperature variation over the dimensions of space and time is requested for a glacier. Following the 12 dynamic modeling steps, which of the following is not correct?

A. An energy balance is needed (Step 5).
Incorrect. An energy balance is needed.
B. Additional thermo relations such as solid heat capacity are needed (Step 6)
Incorrect. Solid heat capacity changes with temperature. These thermo relations are needed for accurate predictions.
C. Spatial dependency indicates that an ordinary differential equation is needed (Step 4)
Correct. Spatial dependency indicates that a partial differential equation is needed (Step 4)
D. There are the same number of variables and equations (Step 7)
Incorrect. For a PDE numerical solution, the ice temperature is divided into control volumes with a temperature for every cell. There is an energy balance for every cell.