Laplace Transforms: Flashcards for Solving Systems of Linear Differential Equations

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Questions and Answers

What is the method used to solve the linear differential equations in the given text?

Laplace transforms

What is the rule used to solve the system of linear algebraic equations in the given text?

Cramer's rule

What is the form of the solution of the linear differential equations obtained using Laplace transforms?

Partial fractions

What is the method used to compute the constants in the solution of the linear differential equations?

Computing the constants using the given equations Signup and view all the answers

What is the final step in obtaining the solution of the linear differential equations?

Taking the inverse Laplace transformations Signup and view all the answers

What is the form of the final solution of the linear differential equations?

Exponential functions Signup and view all the answers

What is the primary use of Laplace transforms in differential equations?

To solve linear differential equations or systems of linear (or linearized) differential equations with constant coefficients. Signup and view all the answers

What is the equation that represents the energy balance in the stirred tank heater at steady-state?

Qv, Ti = Q=UA(Tst-T) Signup and view all the answers

What is the definition of the deviation variable T' in the context of the stirred tank heater problem?

T' = T - Ts Signup and view all the answers

What is the linear differential equation that describes the behavior of the stirred tank heater in terms of the deviation variables?

dT'/dt + aT' = KTs't / τ Signup and view all the answers

What is the purpose of defining deviation variables in the context of the stirred tank heater problem?

To simplify the differential equation by subtracting the steady-state values from the variables. Signup and view all the answers

What is the role of the Laplace transform in solving the linear differential equation that describes the stirred tank heater?

To solve the linear differential equation by transforming it into an algebraic equation that can be easily solved. Signup and view all the answers

What is the most commonly used final control element according to the text?

pneumatic valve Signup and view all the answers

What is the equation that describes the dynamic behavior of inherently second-order processes?

$m \frac{d^2x}{dt^2} + C \frac{dx}{dt} + Kx = PA$ Signup and view all the answers

What is the effect of a proportional control action on the order of the system?

remains the order of the system Signup and view all the answers

What is the force that results from the close contact of the stem with the valve packing?

frictional $dx C dt$ force Signup and view all the answers

What is the characteristic of the stem position in inherently second-order processes?

follows inherent second-order dynamics Signup and view all the answers

What is the effect of an integral control action on the order of the system?

increases the order of the system Signup and view all the answers

What can be a consequence of careless modeling of a process in an input-output model?

A model that does not include all the relevant equations and variables or includes redundant equations and variables. Signup and view all the answers

How does the external world affect the degrees of freedom in an input-output model?

It removes as many degrees of freedom as the number of disturbances. Signup and view all the answers

What is the effect of introducing a control loop in an input-output model?

It removes as many degrees of freedom as the number of control objectives. Signup and view all the answers

What is the purpose of a control loop in the example of a stirred tank heater?

To maintain the temperature of the liquid at the desired value. Signup and view all the answers

What is the equation represented by 𝑡 𝑇𝑠𝑡 = 𝑓(𝑇 = 𝐾𝑐 𝑇 − 𝑇𝑠𝑝 𝑑 𝑇 − 𝑇𝑠𝑝 𝐾𝑐 + ∫ 𝑇 − 𝑇𝑠𝑝 𝑑𝑡 + 𝐾𝑐 𝜏𝐷 + 𝑐𝑠 𝜏𝐼 𝑑𝑡) in the context of the stirred tank heater?

The temperature of the steam. Signup and view all the answers

How are the degrees of freedom reduced in the example of the stirred tank heater?

To 1, by the control loop. Signup and view all the answers

What is the characteristic of a system composed of two noninteracting capacities, where the first system affects the second by its output?

The first system is not affected by the second system, and the solution of the equations is sequential. Signup and view all the answers

In a multi-capacity process, what is the relationship between the capacities and the physical processing unit?

All the capacities can be associated with the same processing unit. Signup and view all the answers

What is the effect of one capacity on the behaviour of the precedent ones in a multi-capacity process?

The effect of one capacity on the behaviour of the precedent ones is referred to as the interacting capacities difference. Signup and view all the answers

What type of dynamics does a system composed of two capacities in series exhibit?

Second-order dynamics. Signup and view all the answers

What is the equation that describes the dynamic behaviour of a stirred tank heater, where Qv0 is the flow rate of the incoming stream and h1 is the hold-up of the tank?

dh1/dt = Qv0 - h1/R1. Signup and view all the answers

What is the significance of the parameters τp1 and τp2 in the differential equations that describe the dynamic behaviour of a system composed of two noninteracting capacities?

They represent the time constants of the first and second systems, respectively. Signup and view all the answers

Study Notes

Solution of Linear Differential Equations using Laplace Transforms

Laplace transforms are used to solve linear differential equations or systems of linear (or linearized) differential equations with constant coefficients.
The method involves rearranging the equations as a linear algebraic system, using Cramer's rule to solve the system, and expanding in partial fractions.
The solution is obtained by taking the inverse transformations.

Example of Linear Differential Equations

A stirred tank heater with constant level is an example of a system that can be solved using Laplace transforms.
The system is described by the equation: dT/dt = UA(Tst - T) / VρCp.
The equation is solved by rearranging it as a linear algebraic system, using Cramer's rule to solve the system, and expanding in partial fractions.

Inherently Second-Order Processes

The pneumatic valve is the most commonly used final control element.
The valve is involved in forces such as pressure exerted by compressed air, force exerted by the spring attached to the stem, and frictional force resulting from the close contact of the stem with the valve packing.
The dynamic behavior of the valve is described by the equation: m d²x/dt² + C dx/dt + Kx = PA.

Dynamic Behavior of Second-Order Systems

Second-order systems can be caused by the presence of controllers in a chemical process.
The type of control action determines the order of the system.
Proportional control remains the order of the system, while derivative control increases the order of the system.

Input-Output Model

The input-output model describes the relationship between the inputs and outputs of a system.
The model can be affected by the presence of controllers and disturbances.
The control system removes as many degrees of freedom as the number of control objectives.

Multicapacity Processes

Multicapacity processes are described by second-order dynamics.
The processes can involve multiple physical processing units or a single processing unit with multiple capacities.
Examples of multicapacity processes include stirred tank heaters and distillation columns.

Solution of Multicapacity Processes

The solution of multicapacity processes involves solving a set of differential equations.
The equations are solved sequentially, as the output of the first system affects the second system.
The solution is obtained by taking the inverse transformations.

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Description

Test your understanding of linear differential equations and their solutions using Laplace transforms. This quiz covers the application of Laplace transforms to solve systems of linear differential equations with constant coefficients, including examples of stirred tank heaters and heat transfer. Brush up on your mathematical skills and get ready to solve some complex equations!

Solving Linear Differential Equations with Laplace Transforms