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Articles by Dr Pankaj Dumka
17 articles
Modelling Steady Flow Energy Equation in Python
Steady Flow Energy Equation (SFEE) is the application of conservation of energy on to an open system. Figure shown below is a schematic of open system in which fluid enters at 𝑖 and exits at 𝑒. The red broken line represents the control surface (CS) of the control volume (CV). The inlet and exit parameters are mentioned in the table shown below − Parameter Inlet Exit Pressure pi pe Velocity Vi Ve Density Pi Pe Specific volume vi ve Enthalpy hi he Area Ai Ae ...
Read MoreModelling Two Dimensional Heat Conduction Problem using Python
In this tutorial, we will see how to model 2D heat conduction equation using Python. A 2D, steady, heat conduction equation with heat generation can be written in Cartesian coordinates as follows − $$\mathrm{\triangledown^{2} T \: + \: \frac{q_{g}}{k} \: = \: \frac{\partial^{2}T}{\partial x^{2}} \: + \: \frac{\partial^{2}T}{\partial y^{2}} \: + \: \frac{q_{g}}{k} \: = \: 0 \:\:\dotso\dotso (1)}$$ This has to be discretized to obtain a finite difference equation. Let us consider a rectangular grid as shown below. The index 𝑖 runs vertically i.e. row wise whereas the index 𝑗 runs horizontally i.e. column wise. Any inside node ...
Read MoreModelling the Taylor Table Method in Python
The Taylor Table method is a very efficient and elegant method of obtaining a finite difference scheme for a particular derivative considering a specific stencil size. To understand it one should be very much clear about what is a stencil. Suppose one wants to evaluate $\mathrm{\frac{d^{2}f}{dx^{2}}}$ then in finite difference method the starting point is the Taylor series. Consider the figure shown below for a better understanding of the method. The Taylor series expansion at the point $\mathrm{x_{i} \: + \: h}$ will be: $$\mathrm{f(x_{i} \: + \: h) \: = \: f(x_{i}) \: + \: hf'(x_{i}) \: + ...
Read MoreModelling Thermodynamic Entropy in Python
Entropy is a property of a thermodynamic system that remains constant during a reversible adiabatic process. Moreover, we can also say that it is the degree of randomness or disorder in the system. If a system exchanges dQ heat from its surroundings at temperature T, then the chance in the entropy can be written as − $$\mathrm{ds \: = \: \frac{dQ}{T} \dotso \dotso \: (1)}$$ According to Clausius' inequality the cyclic integral of $\mathrm{\frac{dQ}{T}}$ along a reversible path is either less or equal to zero. Mathematically, it can be written as − $$\mathrm{\oint\frac{dQ}{T} \: \leq \: 0\dotso \dotso \: (2)}$$ ...
Read MoreModelling the Trapezoidal Rule for Numerical Integration in Python
The purpose of integration (definite) is to calculate the area under a curve of a function between two limits, a and b. The plot shown below will clear this concept further. Quadrature, which is also commonly called as numerical integration, is a method for evaluating the area under the curve of a given function. The process is very simple i.e. first we dividing the bounded area into several regions or strips. Thereafter, the areas is evaluated with the help of mathematical formula of simple rectangle. Then the area of all strips is added to obtain the gross area under ...
Read MoreModelling Stirling and Ericsson Cycles in Python
Stirling Cycle Four processes—two reversible isochoric and two reversible isothermal—make up the Stirling cycle. In the same temperature range, the efficiency of the ideal regenerative Stirling cycle is equivalent to that of the Carnot cycle. Heat interaction takes place throughout the cycle, whereas work interaction only happens in processes 1-2 and 3–4. The figure shown below displays the cycle's schematic. Maximum pressure $\mathrm{(p_{max})}$, minimum pressure $\mathrm{(p_{min})}$, maximum volume $\mathrm{(v_{max})}$, compression ratio (r), and adiabatic exponent $\mathrm{(\gamma)}$ are the input variables taken into consideration when modelling the cycle. The following list includes the thermodynamic calculations of several processes involved in ...
Read MoreModelling the Otto and Diesel Cycles in Python
Otto Cycle An air standard cycle called the Otto Cycle is employed in spark ignition (SI) engines. It comprises of two reversible adiabatic processes and two isochoric processes (constant volume), totaling four processes. When the work interactions take place in reversible adiabatic processes, the heat addition (2-3) and rejection (4-1) occur isochorically (3-4 and 1-2). The Otto cycle's schematic is shown in Figure given below. To model the cycle in Python, the input variables considered are maximum pressure $\mathrm{(P_{max})}$, minimum pressure $\mathrm{(P_{min})}$, maximum volume $\mathrm{(V_{max})}$, compression ratio (r), and adiabatic exponent $\mathrm{(\gamma)}$. Table 2 explains the thermodynamic computations of ...
Read MoreModelling the Gauss Seidel Method in Python
Gauss Seidel Method is the iterative method to solve any system of linear equations. Though the method is very much similar to the Jacobi's method but the values of unknown (x) obtained in an iteration are used in the same iteration in Gauss Seidel whereas, in Jacobi's method they are used in the next iteration level. The updation of x in the same step speeds up the convergence rate. A system of liner equation can be written as − $$\mathrm{a_{1, 1}x_{1} \: + \: a_{1, 2}x_{2} \: + \: \dotso \: + \: a_{1, n}x_{n} \: = \: b_{1}}$$ $$\mathrm{a_{2, ...
Read MoreLumped Capacitance Analysis using Python
When an object at very high temperature is suddenly dropped in a cooler liquid and if it is assumed that the conductive resistance of the solid is very small in comparison to the surrounding convective resistance then the heat transfer analysis is called as lumped capacitance analysis (as shown in the figure given below). Here, we treat the system as a lump. In that case, we can assume that the rate of change of internal energy of lump will be equal to the heat interaction with the surrounding fluid. Mathematically, this can be written as − $$\mathrm{pcV\frac{\partial T}{\partial t} ...
Read MoreImplementation of Jacobi Method to Solve a System of Linear Equations in Python
It is the most straightforward iterative strategy for tackling systems of linear equations shown below. $$\mathrm{a_{1, 1}\: x_{1} \: + \: a_{1, 2} \: x_{2} \: + \: \dotso\dotso \: + \: a_{1, n} \: x_{n} \: = \: b_{1}}$$ $$\mathrm{a_{2, 1} \: x_{1} \: + \: a_{2, 2} \: x_{2} \: + \: \dotso\dotso \: + \: a_{2, n} \: x_{n} \: = \: b_{2}}$$ $$\mathrm{\vdots}$$ $$\mathrm{a_{n, 1} \: x_{1} \: + \: a_{n, 2} \: x_{2} \: + \: \dotso\dotso \: + \: a_{n, n} \: x_{n} \: = \: b_{n}}$$ The fundamental concept is: each linear ...
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