Week 3.5 - Deriving 4th order approximation of a 2nd order derivative using Taylor Table method

Solution 1:   Random Variable number of toys  X Probability  of selling toys P(X) First Business  Moment  `Sigma(x)*P(x)` Second Business Moment `sigma^2=Sigma (x - mu)^2 *P(x)` Third Business Moment `S_k`=`Sigma ((x - mu )/sigma)^3 *P(x)` Fourth Business Moment `K_t = Sigma ((x - mu)/sigma)^4*P(x)…

Question: Why is there a difference in the formula of variance for population and sample?               First of all, we need to understand what we call the population, sample, and variance. So first let's start with the population.               What…

Aim: To perform the combustion simulation of Methane+air on the combustor model and plot the variation of mass fraction of emissions.         To study the effect of adding water to the fuel on emission concentration of species. Introduction:         The combustion simulation can…

Aim: To perform a parametric study on gate valve simulation by setting the opening from 10% to 80%.         To obtain the mass flow rate for each design point.         To calculate the flow coefficient and flow factor for each opening.            Introduction:…

  Aim: To perform analysis on cyclone separator and calculate the separation efficiency and pressure drop. Objective: To perform analysis on a given cyclone separator by varying the velocity from 1 m/s to 5 m/s with particle size 5 micrometer and calculate separation efficiency and pressure drop. To perform analysis…

Aim: To write a code that takes all the reactions from the gri30 mechanism file and gives top 10 most sensitive reaction with respect to temperature. PYTHON PROGRAMME import matplotlib.pyplot as plt import numpy as np import cantera as ct gas = ct.Solution('gri30.yaml') Temp=1500 pre=ct.one_atm gas.TPX=Temp,pre,{'CH4':1,'O2':2…

Aim: To perform CHT analysis on a graphics card with velocity variation 1 m/s to 5 m/s with any three random values.         To find the maximum temperature and heat transfer coefficient attained by the processor.         prove that simulation has achieved convergence with relevant…

Objective -->>1: To observe the effect of pressure and temperature on the auto-ignition time.                1)plotting the auto ignition time for varying pressure from 1 to 5 atm at constant temperature 1250 K.                2)plotting…

Objective: To solve the following coupled ordinary differential equation using python programming by Newton Raphson Method.                 `(dy_1)/(dt)=-0.04*y_1 + (10^4)*y_2*y_3`                 `dy_2/dt=0.04*y_1 - (10^4)*y_2*y_3 -3*(10^7)*y_2^2`…

Stability of the ode    we can say that the ode is stable if small changes in the initial condition of the ode does not affect the final solution of the ode.    1)asymptotically stable          from the figure we can see that no matter what is the value of y at t0 the solution is converging to the same value,than this…

Rayleigh Taylor instability                "It is the instability of dense fluid above a lower density fluid in constant gravitational field is known as rayleigh taylor instability first observed by rayleigh"                 Examples of Rayleigh…

Aim: To derive the reaction rate ODE's and production rate ode for the given reaction with a compact notation so that it can be fed into the computer program easily.   The four reaction are given as                                  `CO + O2 iff CO2 + O` -------------------------------------------------------reaction 1…

 1:What is the effect of preheating temperature on the adiabatic flame temperature?           Adiabatic flame temperature varies linearly with the preheating temperature.              python file  # heat recuparator efficieny calculation """ A furnace…

  WHAT DO YOU MEAN BY CONJUGATE HEAT TRANSFER?? HEAT TRANSFER IN SOLIDS         There are basically three modes of heat transfer named conduction, convection, radiation.         The conduction effect dominates in the heat transfer through solid which is usually due to free electrons…

HANDLING THE MIXTURE WITH CANTERA     In this project, the program is modified and required steps were given in the program itself in form of comments. # handlig mixtures with cantera import cantera as ct gas=ct.Solution('gri30.yaml') #A=ct.Quantity(gas) # we can use general notation for understanding the programme…

Aim-1)Adiabatic flame temperature variation with equivalence ration (constant volume)        2)Adiabatic flame temperature variation with equivalence ration (constant pressure)        3)Adiabatic flame temperature variation with alkane,alkene,alkyne. Introduction Adiabatic Flame…

AHMED BODY The Ahmed body is the generic car body(a simplified vehicle model). The airflow around the Ahmed body captures the essential flow features around the automobile and was first defined and characterized in the experimental work of Ahmed. This model is used to describe the turbulent flow around the car like geometry…

Aim: To simulate the flow around the cylinder to capture the effect of the vortex shedding at the back of the cylinder.        Introduction:                           The flow around the cylinder has opened the path for studying the flow…

Aim -To study the effectiveness of the mixing tee. Simulation conditions steady-state simulation. Mixing effectiveness when hot inlet temperature 36 ° C and cold inlet temperature 19° C. The momentum ratio is equal to 2 and 4. `"Momentum ratio"="velocity of cold inlet"/"velocity of hot inlet"` There are four cases…

Aim=Validation study between symmetric boundary condition, wedge boundary condition vs analytic H.P equation for the flow along the pipe using OPENFOAM. OBJECTIVES To simulate the flow with symmetry and wedge boundary condition for following angles 10,25,45 degrees. To validate hydrodynamic legth with the numerical results.…

Aim-To simulates the flow through the pipe in OpenFOAM. The purpose of this project is to simulate axisymmetric flow by applying the wedge boundary condition. The flow is laminar and flows through a constant cross-section pipe. The Reynolds number is given as 2100. The laminar flow through a constant cross-sectional pipe…

FINITE VOLUME METHOD finite volume method is the method for reducing the partial differential equation into the algebraic equation by dividing domain into finite sized subdomain  through 3 properties conservativeness ,boundness and transportiveness . step1)divide the domain into finite sized subdomain,each subdomain…

Aim-To simulate flow through backward facing step.       -measuring the velocity profile at 8.5m from the inlet.       -To compare the velocity plot the the different grading factors. Steps-   The geometry is than divided into 5 different blocks.  Blockmeshdict in the  tutorials…

Subsonic  Supersonic Isentropic  flow thorough convergent divergent nozzle. Consider the steady ,isentropic flow through convergent divergent nozzle as shown in the figure. The flow enters in the nozzle from the reservoir where the pressure and temperature is denoted by p0 and T0 The cross sectional area is assumed to…

Aim-To create rankine cycle simulator using Matlab. objective -we have three inputs 1)turbine inlet temperature                                                 2)turbine inlet pressure    …

AIM: "To parse NASA thermodynamic data from the text file using matlab code" Objective: To write a code in Matlab that can extract the 14 coefficient from the text file and calculate enthalpy, entropy, specific heat for all the species in the text file.                  To…

Aim-To get the solution of the 2d heat conduction equation. 2d heat conduction equation of Cartesian co-ordinates.   from the figure we can see that we want to compute the value of Tp and our differential equation is elliptic type so we have to  take information from four points around the point Tp    …

What is a genetic algorithm? Basically, it is the technique to optimize the objective function by iteration. starts with the initial generation of candidate solutions and that are tested against the objective function. The key idea is the “evolution is the optimization” It uses the concept of evolutionary biology,…

`(d^2f)/(dx^2)=f''(x)` the number of nodes required for the central difference scheme is=p+q-1 where  p= order of derivative  q= order of approximation so for the central difference approximation the number of node is =p+q-1                          …

%solving the second order differential equation %the system is the damped pendulum clear all close all clc %geometric properties of pendulum b=0.05;%damping factor g=9.81;%gravitational acceleration l=1; %length of pendulum m=1; %mass of the pendulum %initial condition theta_0=[0;3]; %column matrix the first value is displacement…

Aim: To write a programme in octave which can solve otto cycle,create a pv diagram and calculate the efficiency. Introduction to the otto cycle An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition piston engine. It is the thermodynamic…