First Order Systems

Question 1

Simple Examples of FOS: (5)

Answer 1

® Population growth
® Thermometer
® Discharging of a defibrillator
® Radioactive decay
® Renal clearance

Question 2

What are the characteristics of FOS? (3)

Answer 2

• The rate is directly proportional to the level
• The system takes time to response to changes in input unlike immediate response to changes in Zero-Order Systems
• The output is proportional to input no matter how input varies

Question 3

Inflow into a FOS – Model:

Answer 3

Question 4

Inflow into a FOS – Graphical: Exponential Growth Model =

Answer 4

Question 5

Outflow out of a FOS – Model:

Answer 5

Question 6

Outflow out of a FOS – Graphical: Exponential Decay

Answer 6

Question 7

Mathematics of First Order Systems:

What is the solution?

Answer 7

Question 8

Fundamental First Order System Equation:

What does each component of the equation stand for? (6)

Answer 8

L(t) = amount of level after a time interval t
L(0) = initial value of level
e = Exponential function
± = inflow/outflow
k = relative rate of growth/decay
t = Time interval of interest

Question 9

Exponential Decay:

- An outflow from the system which is why there is a -ive exponent in the equation

Answer 9

Question 10

Parameters:

- Decay Rate constant:

Answer 10

• A fractional rate of change in the level
• Symbol = k
• Unit: 1/time OR time-1

Question 11

Parameters: (3)

Answer 11

• rate constant
• time constant
• half-life

Question 12

What are the features of a time constant: (4)

Answer 12

• Characterises the response of the system to a step input
• Symbol: t
• Unit: time
• Inverse of rate constant t = 1/k

Question 13

What are the features of half-life? (3)

Answer 13

• Time taken for level to decrease to half of its original amount
• Symbol: t1/2
• Unit: time

Question 14

Exponential Decay shown graphically:

Answer 14

Question 15

Recovery shown graphically:

Answer 15

Question 16

What is exponential growth?

Answer 16

—-> an inflow into the system which is why there is a +’ve exponent in the equation

Question 17

Parameters for exponential growth: (3)

Answer 17

• Growth Rate constant
• Time constant
• Doubling time

Question 18

What are the features of the Growth Rate constant?

Answer 18

• A fractional rate of change in the level
• Symbol = k
• Unit: 1/time OR time-1

Question 19

What are the features of the time constant? (4)

Answer 19

• Characterises the response of the system to a step input
• Symbol: t
• Unit: time
• Inverse of rate constant t = 1/k

Question 20

What are the features of doubling time? (3)

Answer 20

• Time taken for level to increase to double its original amount
• Symbol: t2
• Unit: time

Question 21

Exponential Growth shown graphically:

Answer 21

Question 22

Example 1:
Approximate the process of water emptying out of a bathtub under the influence of gravity as a first-order process with a rate constant of 0.1 min-1. Assuming that there are initially 200 litres of water in the bath, how many minutes will it take for the water in the bathtub to decrease to 10% of the original volume?

Given:
L(0) = 200L
First order system
k = 0.1min-1

Answer 22

Question 23

Example 2:
There are initially 150 litres of water in a tank when the plug is pulled out of the drain and the tank empties into a second, lower tank, with a time constant of 10 minutes. Calculate how long it will take in minutes for the volume of water in the lower tank to just exceed the volume in the upper tank?

Given:
L1(0) [initial level in first tank] = 150L
t = 10 minutes
Assume L2(0) = 0L

Answer 23