First Order Systems Flashcards
Simple Examples of FOS: (5)
® Population growth ® Thermometer ® Discharging of a defibrillator ® Radioactive decay ® Renal clearance
What are the characteristics of FOS? (3)
- 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
Inflow into a FOS – Model:
Inflow into a FOS – Graphical: Exponential Growth Model =
Outflow out of a FOS – Model:
Outflow out of a FOS – Graphical: Exponential Decay
Mathematics of First Order Systems:
What is the solution?
Fundamental First Order System Equation:
What does each component of the equation stand for? (6)
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
Exponential Decay:
- An outflow from the system which is why there is a -ive exponent in the equation
Parameters:
- Decay Rate constant:
- A fractional rate of change in the level
- Symbol = k
- Unit: 1/time OR time-1
Parameters: (3)
- rate constant
- time constant
- half-life
What are the features of a time constant: (4)
- Characterises the response of the system to a step input
- Symbol: t
- Unit: time
- Inverse of rate constant t = 1/k
What are the features of half-life? (3)
- Time taken for level to decrease to half of its original amount
- Symbol: t1/2
- Unit: time
Exponential Decay shown graphically:
Recovery shown graphically:
What is exponential growth?
—-> an inflow into the system which is why there is a +’ve exponent in the equation
What are the features of the Growth Rate constant?
- A fractional rate of change in the level
- Symbol = k
- Unit: 1/time OR time-1
What are the features of the time constant? (4)
- Characterises the response of the system to a step input
- Symbol: t
- Unit: time
- Inverse of rate constant t = 1/k
What are the features of doubling time? (3)
- Time taken for level to increase to double its original amount
- Symbol: t2
- Unit: time
Exponential Growth shown graphically:
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
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
