Simple ODEs to Study Thermofluids P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Understanding of A Single DoF System ….

The Laplacian Operator In Cartesian Coordinates In Cylindrical Coordinates In Spherical Coordinates

Elementary potential flows-1 One dimensional Laplacian flow in Spherical Coordinate system: where m and C are integration constants.

Application in Hydrodynamics This potential is considered as velocity potential, it produces the three-dimensional radial velocity. m is called as fluid strength. If m > 0 this is called as a point source or point sink, if m < 0. By convention the constant m = Q/4 where Q is the flow rate or volume flux.

Application in Conduction This potential is considered as Temperature, it describes radial heat conduction in a spherical solid. Boundary conditions: Dirichlet conditions : Temperatures specified at r0, r1, r2 and r3. Neumann conditions t conditions : T/r specified at r0, r1, r2 and r3.

Elementary Vector Calculus Functions for Thermofluids Solenoidal Flow Fields : Irrotational (Lamellar) Flow Fields : Potential Flow Fields, Steady state conduction & Steady state mass diffusion : Beltrami Flow Fields : Complex Lamellar Field

Assignment 1 Submit a research report on: Two applications of Beltrami Flows. Two applications of Complex Lamellar Flows. Date of Submission: 20th August 2018. Original Reports will be given highest marks. All copied will be given zero marks.

Vectorially Described Thermofluid Fields

It is Essential to know Solutions of ODEs to Solve True Thermofluid Mathematical Models

First Order Differential Equations for Thermofluids The general first-order differential equation for the function y = y(x) is written as where f (x, y) can be any function of the independent variable x and the dependent variable y. It is not always possible to find an analytical solution.

Separable First Order Differential Equations A first-order ode is separable if it can be written in the form where the function v(y) is independent of x and u(x) is independent of y. Homogeneous first order ODE: Inhomogeneous first order ODE:

Solution of Separable Homogeneous First ODE Divide through by v(y) to obtain Proceed to integrate both sides of this equation with respect to x, to get Variables are separated, because the left-hand side contains only the variable y and the right-hand side contains only the variable x. It can be tried to integrate each side separately. If required integration is actually performed, it is possible to obtain a relationship between y and x.

Design of Thermometers in Thermofluids Zeroth Law of Thermodynamics Every system in this universe spontaneously move towards equilibrium with surroundings. If A and C are in thermal equilibrium with B, then A is in thermal equilibrium with C. Maxwell [1872] Design Rule: Every thermal instrument must be designed to reach thermal equilibrium with the system as fast as possible. Independent variable : time t Dependent variable : Temperature T Engineering Mathematics: Obtain a suitable ODE for this application.

Construction of A Thermometer

Options for Thermometers Liquid in Glass Thermometer Thermocouple Thermometer

Design Criteria : How Long it takes to achieve Zeroth Law? Conservation of Energy during a time dt Heat in = Change in energy of bulb = Instantaneous Temperature of the bulb

Governing ODE for Thermometer Bulb Define Time constant

Design for Constant System Temperature Define relative temperature of the bulb as Initial condition Equilibrium condition

Instantaneous Relative Temperature of Bulb Initial condition

Response of A Thermometer bulb in A Constant Temperature System

Need for Inhomogeneous First Order ODE Ts(t)=Ct

Linear Inhomogeneous First Order ODEs The first-order linear differential equation (linear in y and its derivative) can be written in the form with the initial condition y(x0) = y0. p(x) and g(x) are functions of x only. It is defined as a linear equation, as each term involves y either as the derivative dy/dx OR through a single factor of y .

Theory of Solution A linear first order ODE can be solved using the integrating factor method. The equation may be multiplied by the “Integrating Factor”, (x). Multiply the original ODE with an integrating factor (x). The IF is defined so that the equation becomes equivalent to: