Steady-State Help: Solve Equations & Find x0 & u0

  • Thread starter bobboviking
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Your Name]In summary, the conversation revolves around finding the values of x0 and u0 for steady-state evaporator equations. The equations involve constants B1, B2, B3, Ta, Ltot, and hg, as well as state variables x and control inputs u. Two methods, Newton's method and numerical solver, can be used to solve the equations by making an initial guess and iterating until a solution is found. It is important to have a reasonable initial guess for the values of x and u.
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bobboviking
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Steady-state HELP!

Hi guys, I really need some help with this!

I have this set of equations for an evaporator which I want to calculate x0 and u0 for the steady state

mdotin(hin - hg) + (B1/Ltot)*L1(Tw1 - Tr1) = 0
mdotout(hg - hout) + (B2/Ltot)*L2(Tw2 - Tr2) = 0
mdotin - mdotout = 0
B3(Ta - Tw1) -B1(Tw1 - Tr1)
B3(Ta - Tw2) -B2(Tw2 - Tr2)

where B1,2,3 are constant values
Ta is the surrounding air on the evaporator
Ltot is the total length of the evaporator
L2 = Ltot - L1
hg the enthalpy of the vapor refrigerantMy state variables are x = [L1 P hout Tw1 Tw2] and my control inputs are u = [mdotin mdotout hin] which results in the following set of equations

u1(u3 - hg) + (B1/Ltot)*x1(x4 - Tr1) = 0
u2(hg - x3) + (B2/Ltot)*(Ltot - x1)(x5 - Tr2) = 0
u1 - u2 = 0
B3(Ta - x4) - B1(x4 - Tr1) = 0
B3(Ta - x5) - B2(x5 - Tr2) = 0I don't really know how to find the values of x0 and u0 from here.
I would highly appreciate some help!

Thanks in advance,
Andreas
 
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Dear Andreas,

Thank you for reaching out for help with your steady-state evaporator equations. I would be happy to assist you with finding the values for x0 and u0.

To solve for the steady state, we need to find the values of x and u that satisfy the equations you have listed. This can be done through a process called numerical simulation, where we use mathematical methods to approximate the solutions.

First, we need to define the specific values for the constants B1, B2, B3, Ta, Ltot, and hg. These values can come from your experimental setup or can be estimated based on previous studies. Once we have these values, we can start the numerical simulation.

One approach to solving these equations is to use a method called "Newton's method". This method involves making an initial guess for the values of x and u, and then using the equations to update these values until we reach a solution that satisfies all the equations. This process is repeated until the values of x and u converge to a steady state.

Another approach is to use a numerical solver, such as the "fsolve" function in MATLAB, which can solve systems of nonlinear equations. This method also involves making an initial guess for the values of x and u, and then the solver will iterate until it finds a solution that satisfies all the equations.

In both cases, it is important to make a reasonable initial guess for the values of x and u. This can be done by using your knowledge of the system and its behavior, or by using values from previous experiments.

I hope this helps you in finding the values of x0 and u0 for your steady-state evaporator equations. If you need further assistance, please don't hesitate to reach out.
 

1. What is steady-state?

Steady-state refers to a condition in a system where all variables remain constant over time. This means that the system is in a stable equilibrium and there is no net change in any of its components.

2. How do you solve equations to find x0 and u0 in steady-state?

To solve equations for steady-state, you need to set the derivatives of all variables to zero. This means that the system has reached a point where all variables are constant and there is no change over time. From there, you can solve for the values of x0 and u0.

3. What is the importance of finding x0 and u0 in steady-state?

Finding x0 and u0 in steady-state allows us to understand the behavior of a system over time and predict its future outcomes. It also helps us to identify any potential issues or instabilities in the system that may arise.

4. What are the common methods used to solve steady-state equations?

The most common methods used to solve steady-state equations are the substitution method, elimination method, and graphing method. These methods involve manipulating the equations to eliminate variables and solve for the unknowns.

5. How is steady-state different from dynamic equilibrium?

Steady-state and dynamic equilibrium are both conditions of stability in a system. However, steady-state refers to a constant state where all variables remain unchanged over time, while dynamic equilibrium refers to a system that is constantly changing but maintains a stable overall state.

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