## Quiz on Transfer Functions

**Information for Questions 1-3**: Two reactors are operated in parallel and the output of each is mixed in a blending tank. The inputs (`U_1` and `U_2`) are cooling jacket temperatures. The reactor outputs (`X_1` and `X_2`) are reactor concentrations that are mixed (`X_3`) and blended (`Y`).

**1.** What is the expression for reactor outlet `X_2` in terms of input `U_2`?

**A.**`X_2 = X_1 U_1`

- Incorrect. Use the transfer function `G_2` and input `U_2`

**B.**`X_2 = G_1 U_2`

- Incorrect. Use the transfer function `G_2`

**C.**`X_2 = G_2 U_2`

- Correct. The transfer function `G_2` relates the input `U_2` to the output `X_2`

**D.**`X_2 = G_1 U_1`

- Incorrect. Use the transfer function `G_2` and input `U_2`

**2.** What is the expression for the mixed reactor outlet `X_3` in terms of inputs `U_1` and `U_2`?

**A.**`X_3 = G_1 G_2 U_1 U_2`

- Incorrect. Processes in parallel are additive, not multiplicative

**B.**`X_3 = X_1 U_1 + X_2 U_2`

- Incorrect. Use transfer functions `G_1` and `G_2` instead of `X_1` and `X_2`

**C.**`X_3 = X_1 + X_2`

- Incorrect. This is correct but it is not in terms of the inputs

**D.**`X_3 = G_1 U_1 + G_2 U_2`

- Correct. `X_3 = X_1 + X_2 = G_1 U_1 + G_2 U_2`

**3.** What is the expression for the blender outlet `Y` in terms of inputs `U_1` and `U_2`?

**A.**`Y = G_3 X_3`

- Incorrect. This is correct but not in terms of `U_1` and `U_2`

**B.**`Y = G_3 (X_1 + X_2)`

- Incorrect. This is correct but not in terms of `U_1` and `U_2`

**C.**`Y = G_3 G_1 U_1 + G_3 G_2 U_2`

- Correct. This is in terms of the inputs `U_1` and `U_2` and the transfer functions

**D.**`Y = G_3 X_3 G_1 X_1 U_1 + G_3 X_3 G_2 X_2 U_2`

- Incorrect. The intermediate signals `X_1`, `X_2`, and `X_3` are not in the input to output expression

**4.** What are the requirements to obtain a transfer function? Select two correct answers.

**A.**Nonlinear equations

- Incorrect. Only linear equations or linearized nonlinear equations can be transformed to Laplace domain

**B.**Linear equations

- Correct. Only linear equations can be transformed to Laplace domain

**C.**Differential equation with multiple inputs and one output

- Incorrect. A transfer function relates one input to one output

**D.**Differential equation with one input and one output

- Correct. A transfer function relates one input to one output

**5.** What is the gain of the transfer function `G(s) = {5(s + 1)}/{(s + 3)^2}`?

**A.**5/3

- Incorrect. `K_p = \lim_{s \to 0} {5(s + 1)}/{(s + 3)^2}`

**B.**5/9

- Correct. `K_p = \lim_{s \to 0} {5(s + 1)}/{(s + 3)^2} = {5(1)}/{3^2} = 5/9`

**C.**0

- Incorrect. `K_p = \lim_{s \to 0} {5(s + 1)}/{(s + 3)^2}`

**D.**1

- Incorrect. `K_p = \lim_{s \to 0} {5(s + 1)}/{(s + 3)^2}`