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Mathematical Time Constant |
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There are a number of real-world physical systems whose functionality and dynamic response characteristics can be mathematically modeled using a relatively simple first-order linear differential equation of the form 
. Demonstrating the effect of the Time Constant characteristic of such systems on their dynamic response is generally accomplished by implementing a step change to one of the system’s input parameters rather than by using a forcing/driving function (
), so that the general solution to this differential equation (with the forcing/driving function 
) is of the form 
, where 
is the dynamic (time-dependent) response of the system, 
is the value of the response function at time 
, and 
(the Greek letter tau) is the Time Constant characteristic of the system.
For example, the differential equation that models a well-mixed tank containing a solution of salt in water is given by 
while its response solution is given by 
where the Time Constant of the system () is 10 minutes. (Click here to see the details of the Well-Mixed Tank Example.)
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RC Electrical Circuit Example
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RL Electrical Circuit Example
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Well-Mixed Tank Example
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A tank has pure water flowing into it at the rate of 10 gallons per minute. The contents of the tank are kept thoroughly mixed, and the contents flow out of the tank at the rate of 10 gallons per minute. Initially, the tank contains 10 pounds of salt in 100 gallons of water. There is negligible effect of the dissolved salt on the volume of the solution. How much salt will there be in the tank after 30 minutes? |
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Mathematical Analysis/Modeling and Discussion
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Graphical Response Representation
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Well-Mixed Tank Example
Alternate Solution
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A tank has pure water flowing into it at the rate of 10 gallons per minute. The contents of the tank are kept thoroughly mixed, and the contents flow out of the tank at the rate of 10 gallons per minute. Initially, the tank contains 10 pounds of salt in 100 gallons of water. There is negligible effect of the dissolved salt on the volume of the solution. How much salt will there be in the tank after 30 minutes? |
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Mathematical Analysis/Modeling and Discussion
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Graphical Response Representation
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Heat Transfer (Heating) Example
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Heat Transfer (Cooling) Example
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