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D-4536-2 1 Beginner Modeling Exercises Section 3 Mental Simulation of Simple Negative Feedback Stock Flow Decay Factor Prepared for MIT System Dynamics in Education Project Under the Supervision of Dr. Jay W. Forrester by Helen Zhu Vensim Examples added October 2001 Copyright © 2001 by the Massachusetts Institute of Technology Permission granted to copy for non-commercial educational purposesD-4536-2 3 Table of Contents INTRODUCTION 5 NEGATIVE FEEDBACK EXAMPLE 1: RAINFALL ON THE SIDEWALK (GOAL = 0) 6 THE SYSTEM AND ITS INITIAL BEHAVIOR 6 HALF-LIFE CALCULATIONS FOR RAINFALL SYSTEM 9 EXAMPLE 2: CHROMATOGRAPHY (GOAL ? 0) 10 REVIEW 14 EXPLORATION SOLUTIONS BIBLIOGRAPHY VENSIM EXAMPLES 5 14 19 26 27D-4536-2 5 Introduction Feedback loops are the basic structural elements of systems. Feedback in systems causes nearly all dynamic behavior. To use system dynamics successfully as a learning tool one must understand the effects of feedback loops on dynamic systems. One way of using system dynamics to understand feedback is with simulation software on your computer1. Computer simulation is a very useful tool for exploring systems. However, one should be able to use the other simulation tool of system dynamics: mental simulation. A strong set of mental simulation skills will enhance ability to validate, debug and understand dynamic systems and models. This paper deals primarily with negative feedback and begins with a review of some key concepts. A set of exercises at the end will help reinforce understanding of the feedback dynamics in a simple negative feedback loop. Solutions to these exercises are also included. It is assumed that the reader has already studied “Beginner Modeling Exercises: Mental Simulation of Positive Feedback,”2 and is familiar with fundamental system dynamics terms. Negative Feedback Compare negative feedback to letting air out of a balloon. At first, air pressure inside the expanded balloon pushes air out at a high rate, allowing the balloon to deflate. As air escapes, the balloon gets smaller, the air pressure dies down, and the deflation rate drops. This continues until deflation stops completely. Negative feedback occurs when change in a system produces less and less change in the same direction until a goal is reached. In this circumstance, the goal is equal air pressure inside and outside the balloon. 1There are several commercial system dynamics simulation packages available for both Windows and Macintosh. Road Maps is geared towards the use of STELLA II which is available from High Performance Systems (603) 643-9636. Road Maps can be accessed through the internet at http:/sysdyn.mit.edu/.2 “Beginner Modeling Exercises: Mental Simulation of Positive Feedback” (D-4487) by Joseph G. Whelan, is part of the Road Maps series.6 D-4347-7 Systems exhibiting negative feedback are present everywhere, ranging from a population facing extinction to the simple thermostat. In each circumstance, negative feedback exhibits goal-seeking behavior. In other words, the difference between the current state of the system and the desired state causes the system to move towards the desired state. The closer the state of the system is to its goal, the smaller the rate of change, until the system reaches its goal. This goal can differ from system to system. In a population extinction model, the goal is zero — animals continue dying until there are none left. A thermostat, on the other hand, has a goal of a desired room temperature. Example 1: Rainfall on the sidewalk (Goal = 0) This section focuses on how to simulate and visualize stock behavior mentally over time for a zero-goal system. The first step explores initial behavior. The system and its initial behavior An example of negative feedback is the amount of wet sidewalk from rain. Initially, a 10 square foot block of sidewalk is completely dry. Rain begins to fall at a constant rate. Within the first minute, rain covers 50% of the block surface area, or 5 square feet total. In the next minute, the same amount of rain falls, but some of the raindrops fall on areas that are already wet. The chance of rain falling on a dry spot is one out of two, or 50%. Rainfall covers five square feet in the second minute, but 50% of that is already wet, so only an additional 2.5 square feet becomes wet. wet and dry dry wetD-4536-2 7 Figure 1: Rain falls on sidewalk, first minute Although rainfall is evenly distributed in “wet and dry,” it quantitatively covers half of the surface area of the block. It is easier to visualize this, for modeling purposes, as its equivalent of one-half wet and one-half dry. Figure 1 above shows both the natural rainfall pattern and its equivalent schematic representation. Rain continues to fall. Half of it lands on places where the sidewalk is already wet. The other half covers 50% of the remaining dry areas. wet wet and dry dry wet Figure 2: Sidewalk, minute two Net rainfall is the equivalent of 75% wet, 25% dry. The same process of 50% of the remaining dry surface becoming wet continues, advancing towards covering the entire surface area. wet wet and dry wet dry Figure 3: Sidewalk, minute three Given this description of the rainfall system, we can model it in STELLA as shown in the figure below.8 D-4347-7 dry surface area rain coverage coverage fraction Figure 4: STELLA model of rainfall system rain coverage = dry surface area * coverage fraction [square feet/min] [square feet] [/minute] We know that: initial dry surface area = 10 square feet coverage fraction = 1/2 /minute Thus, initial rain coverage = 10 * 1/2 = 5 square feet/minute. In other words, as Figure 5 shows, the initial slope of the dry surface area is -5 square feet per minute. This negative term indicates that dry surface area decreases by rain coverage rate.D-4536-2 9 1: dry surface area 10.00 1 Initial dry Initial slsope = -5 square feet/minute urface area = 10 square feet square feet5.00 0.00 0.00 2.00 4.00 6.00 8.00 minutes Figure 5: Behavior plot of rainfall system What happens past the initial point? Eventually rain will cover the entire sidewalk. The system moves toward the goal of zero dry surface area. In order to mentally simulate the quantitative behavior of the system over time, one should perform half-life calculations. Half-life Calculations for rainfall system Negative feedback systems exhibit asymptotic behavior. Asymptotic decay of a stock has a constant halving time, or


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MIT 15 988 - Beginner Modeling Exercises

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