laine

laine solves nonlinear equations using the simultaneous solution approach. It is meant to be intuitive and requires no background in computer programming. Try it here!.

Quick start

1. Write a set of equations which represents the problem. For instance:

    x*y=3 
    x^2+y^2 = 10
   

2. Click Solve.

3. The solution should appear on the screen.

    x = 1
    y = 3
   

That's it. You just have to write equations and laine should be able to solve them for you.

Syntax rules

An equation should be written as: left expression = right expression. For example:

T = 25+273.15

The same equation could have been written as:

25+273.15 = T

Thus, you can write equations in the most convenient way for you.

Operation symbols

Operation	Symbol
Sum	+
Subtraction	-
Multiplication	*
Division	/
Power	^

Some conventions:

• laine uses dot ( . ) as a decimal separator.

    half  = 0.5
  

• Strings should be input between pairs of quotation marks ( " or ' ).

    Variable = "word"
  

• If you want to insert more than one equation in a line, just separate them by using a semicolon ( ; ).

    T = 300 # K ; P = 101325 # Pa
  

• (In Reports) Variables with Greek letters can be inserted with an dollar sign ($).

    $DeltaV = 30 # m³
  

• (In Reports) If is just a Greek letter, than it is not necessary to put a dollar sign:

    rho_water = 999 # kg/m³
  

Using functions

laine uses math.js to parse equations, therefore it inherited the math function of this library. In the following table there are some examples, the complete list can be seen here.

Function	Syntax
exponent	exp(x)
logarithm (base e)	log(x)
logarithm (base 10)	log10(x)
cosine	cos(x)
sine	sin(x)
tangent	tan(x)

Furthermore, laine also has some thermodynamic functions that you can access in Functions:

• Pure substances (PropsSI): thermodynamic properties for pure substances using the CoolProp library.

• Humid air (HAPropsSI): humid air properties also using the CoolProp library.

• Reactions (NasaSI): properties of some species in the standard state using Nasa Gleen coefficients.

• Lee-Kesler (LeeKesler): the deviation from ideal gas properties for simple fluids. Based on the original publication by Lee and Kesler (1975).

User-defined functions

laine also supports user-defined functions by using the syntax: function(var1,var2,...) = expression. For example:

T_kelvin(T_celsius) = T_celsius+273.15

Comments

Since it is easy forget what an equation means, you can input a comment by inserting '#'. Unfortunately laine does not have a multiline comment option (yet).

# Environmental pressure
P = 101325 # Pa 

In the 'Report' mode, laine renders the comments and equations as markdown and Latex using showdown and MathJax, respectivelly. Thus, you can write sections, emphasis and LaTeX (equations) if necessary.

    ## Header 1
    ### Header 2
    # This is an inline LaTeX equation \(W = \int P dv \), here markdown is disabled
    # $$ a^2 = b^2 + c^2 $$
    # Here is a _markdown emphasis_

However, we disabled markdown rendering in lines that you use equations to avoid errors.

User guesses

In some cases, laine requires that the user provides an initial guess for a certain variable. An initial guess can be declared like equations but ending with a question mark (?). For example:

    x = 3 ?

This helps laine to converge faster or to find a specific solution.

Parametric analysis

A problem consists in a set of equations, which can only be solved for a unique solution if the number of equations is equal to the number of variables.

Differently from solving a problem, a Parametric Analysis varies a certain variable (x), records the solution for another variable (y) and plot the results. Thus, a Parametric analysis requires that the set of equations has one degree of freedom. For instance:

y = 2*x

In this problem, there are 2 variables (y,x) and one equation. Then you can vary one of the two variables (e.g., x) and plot the results (e.g., y vs. x).

Property plots

In this option, laine scans the code for calls of the function PropsSI and returns a list of states that can be plotted in a diagram of thermodynamic properties (e.g., T-s, P-h, P-v).

laine automatically checks if sequential states have a certain property in common and draws the process accordingly. For instance, an isentropic process in a P-v diagram is represented by a curve of isentropic points and not a simple straight line.