# Need Help Solving Set of Coupled ODEs

• Stanley Park
In summary, the conversation discusses the process of solidifying liquid nitrogen by connecting it to a vacuum pump. As the gas is removed, the pressure in the dewar decreases, causing the evaporation rate to increase and the temperature to decrease due to latent heat. The goal is to set up a model to calculate the time for the solidification process. Several equations are mentioned, including the evaporation rate, temperature change, vapor pressure change, and time calculation. However, combining these equations may be difficult and it is suggested to use LaTex for better understanding. The thread has been moved to a math forum for further assistance.
Stanley Park

## Homework Statement

Liquid nitrogen is in a dewar connected to a vacuum pump. Initial pressure in a dewar is 1atm and saturated with gaseous nitrogen. If the vacuum pump started, it removes gas in it and the pressure in a dewar will be reduced under the saturation pressure of the liquid nitrogen. Then evaporation rate will be increased and the temperature of the liquid nitrogen will be decreased due to the latent heat. And when the temperature of the nitrogen closed the phase change temperature, liquid nitrogen will be solidified. I have to set up a model to calculate the time for the solidification.

## Homework Equations

1) The evaporation rate can be defined as m dot = hA(Pv-Ph) where h is boiling coefficient, Pv and Ph is saturation pressure and environment pressure respectively.
2) Temperature change of liquid nitrogen : cm_ln*dT/dt = m dot*hfg where c is specific heat, m_ln is mass of liquid nitrogen, hfg is latent heat.
3) Vapor pressure change : dPv/dt = -m dot*R*Tk/(Mn2*Vf) where, Tk is the temperature of a dewar, Mn2 is nitrogen's molecular weight and Vf is free volume in a dewar.
4) Time to reach P=p2 from p1 : tp=(V/Ss)ln[(p1-pu)/(p2-pu)] where Ss is system pumping speed, pu is the ultimate pressure.

## The Attempt at a Solution

I have hard time combining upper mentioned equations. Also it is to complicate to solve.

Please read the tutorial on LaTex text editor and rewrite you equations using LaTex. As the equations are written, they are very hard to understand mathematically. Also, I have moved your thread to a math forum where you may get more of a response.

## 1. What is a set of coupled ODEs?

A set of coupled ODEs (ordinary differential equations) is a system of two or more differential equations that are interdependent and cannot be solved independently. This means that the solution to one equation is dependent on the solutions of the other equations in the system.

## 2. How do I know if a set of coupled ODEs can be solved analytically?

Analytically solving a set of coupled ODEs can be challenging and not all systems have an analytical solution. In general, if the equations in the system are linear and have constant coefficients, there is a higher chance of finding an analytical solution. However, for nonlinear systems or systems with time-varying coefficients, numerical methods may be necessary.

## 3. What are some common methods for solving a set of coupled ODEs?

Some common methods for solving a set of coupled ODEs include numerical methods such as Euler's method, Runge-Kutta methods, and finite difference methods. These methods involve discretizing the equations and finding approximate solutions at discrete time intervals. Other methods include Laplace transforms and separation of variables for simpler systems.

## 4. How do I choose the appropriate method for solving a set of coupled ODEs?

The choice of method depends on the specific characteristics of the system, such as linearity, number of equations, and initial conditions. For simpler systems, analytical methods may be preferred, but for more complex systems, numerical methods may be necessary. It is important to consider the trade-offs between accuracy, computational cost, and ease of implementation when choosing a method.

## 5. Are there any software programs or tools that can help with solving a set of coupled ODEs?

Yes, there are many software programs and tools available for solving sets of coupled ODEs. Some popular options include MATLAB, Mathematica, and Python libraries such as SciPy. These programs offer various numerical methods and allow for easy visualization and analysis of the solutions. Additionally, there are online ODE solvers and tutorials available for those without access to specialized software.

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