Q1: A bar of length 1 m is insulated along its sides. It is initially at a uniform temperature of 15° C and a t=0 the ends are plunged into ice and maintained at a temperature of 0°C. 1. State all Initial and boundary conditions 2. Determine an expression for the temperature distribution at a point P a distance x from one end at any subsequent time t seconds after t=0, 3. write all the Eigen values and Eigen function possible a2u 1 ди - ax2 c2 at

Q1: A bar of length 1 m is insulated along its sides. It is initially at a uniform temperature of 15° C and a t=0 the ends are plunged into ice and maintained at a temperature of 0°C. 1. State all Initial and boundary conditions 2. Determine an expression for the temperature distribution at a point P a distance x from one end at any subsequent time t seconds after t=0, 3. write all the Eigen values and Eigen function possible a2u 1 ди - ax2 c2 at

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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A bar of length 1 m is insulated along its sides. It is initially at a uniform temperature of 15°C and a t=0 the ends are plunged into ice and maintained at a temperature of 0°C. 1. State all Initial and boundary conditions 2. Determine an expression for the temperature distribution at a point P a distance x from one end at any subsequent time t seconds after t-0, 3. write all the Eigen values and Eigen function possible 1 ди %3D dx? c2 at
Question 2: Air at the temperature of T1 is being heated with the help of a cylindrical cooling fin shown in the figure below. A hot fluid at temperature To passes through the pipe with radius Rc. a) Derive the mathematical model that gives the variation of the temperature inside the fin at the dynamic conditions. b) Determine the initial and boundary conditions to solve the equation derived in (a). air T1 Re Assumptions: 1. Temperature is a function of onlyr direction 2. There is no heat loss from the surface of A 3. Convective heat transfer coefficient is constant
Question 2: Air at the temperature of T1 is being heated with the help of a cylindrical cooling fin shown in the figure below. A hot fluid at temperature To passes through the pipe with radius Rc. a) Derive the mathematical model that gives the variation of the temperature inside the fin at the dynamic conditions. b) Determine the initial and boundary conditions to solve the equation derived in (a). air To Re Assumptions: 1. Temperature is a function of only r direction 2. There is no heat loss from the surface of A 3. Convective heat transfer coefficient is constant
Differential equation THE LAPLACE METHOD CANNOT BE USED. When two springs in series, with constants k1 and k2 respectively,support a mass, the effective spring constant is calculated as:k = k1*k2 / k1+k2 A 420 g object stretches a spring 7 cm, and that same object stretches 2.8 cmanother spring. Both springs are attached to a common rigid support and then toa metal plate, as shown in the figure of springs in series. Then the object joins thecenter of the plate (figure of springs in series). Determine: (a) the effective spring constant.(b) the position of the object at any time t, if the object is initially releasedfrom a point 60 cm below the position ofequilibrium and with an upward velocity of 1.2 m / s.Consider the acceleration of gravity as 9.8 m / s2.
differential equation A metal bar is placed in a room at constant temperature of 0°C. If after 25 mins the temperature of bar drops from 120°C to 55°C, find how many minutes the metal bar will continue to cool from 55°C to 50°C.
The wall (thickness L) of a furnace is comprised of brick material (thermal conductivity, k = 0.2 Wm¯' K'). Given that the atmospheric temperature is 0°C at both sides of wall, the density (p) and heat capacity (c) of the brick material are 1.6 gm cm³ and 5.0 J kg K¯l respectively. du Solve pc = k- subject to initial conditions as u(x,0) = x²(L – x). ốt Consider the case 2=- p² only.
5 Problem = How long should it take to boil an egg? Model the egg as a sphere with radius of 2.3 cm that has properties similar to water with a density of ρ 1000 kg/m³ and thermal conductivity of k = 0.606 Watts/(m.°C) and specific heat of c = 4182 J/(kg.° C). Suppose that an egg is fully cooked when the temperature at the center reaches 70° C. Initially the egg is taken out of the fridge at 4° C and placed in the boiling water at 100° C. Since the egg shell is very thin assume that it quickly reaches a temperature of 100° C. The protein in the egg effectively immobilizes the water so the heat conduction is purely conduction (no convection). Plot the temperature of the egg over time and use the data tooltip in MATLAB to make your conclusion on the time it takes to cook the egg in minutes. Figure 1: Image source: [Link]
The temperature on a sheet of metal is known to vary according to the following function: T(z,9) – 4z - 2ry We are interested to find the maximum temperature at the intersection of this sheet with a cylindrical pipe of negligible thickness. The equation of the intersection curve can be approximated as: 2+-4 Find the coordinates for the location of maximum temperature, the Lagrangian multiplier and the value of temperature at the optimum point. Wnite your answer with two decimal places of accuracy. HINT: IF there are more than one critical point, you can use substitution in the objective function to select the maximum. Enter your results here: Optimum value of z Optimum value of y Optimum value of A Optimum value of T
10. From Newton's law of cooling, the boiled egg submersed in water in the diagram below will cool according to dI = -0. 1(T – a), T(0) = 100°C . Solve this differential equation and express T in terms of a. Do not use the integrating factor method. Present your solution with T as the subject. Use your solution to solve the following: If after 1 = 10 time units this egg cools to T = 2ª°C , what was the value of a (a is the temperature of the water in the glass which is assumed to have remained constant)?
Consider the following for a single-degree-of-freedom system with m = 1. X, k = 2.5, and c = 1.8. Here, the value of X indicating of the number of your Q1 (a) group. For example, if your group number is 15 gives X = 15, therefore the values of m = 1.15. If your group number is 9 gives X = 9, therefore the values of m = 1.9 Find; Natural frequency, Wn (ii) Linear frequency, fn (iii) Critical damping constant, c. (iv) Damping ratio, 3 (v) Damped frequency, Wa
1. A spring mass system serving as a shock absorber under a car's suspension, supports the M=1000kgmass of the car. For this shock absorber,k=1000N/m and c=2000N s/m. The car drives over a corrugated road with force F=2000sin(wt)N. Use your notes to model the second order differential equation suited to thisapplication. Simplify the equation with the coefficient of x'' as one. Solve x (the general solution) interms of using the complimentary and particular solution method. In determining the coefficients ofyour particular solution, it will be required that you assume w2 -1=w or . Do not 1-w2=-wuse Matlab as its solution will not be identifiable in the solution entry. Do not determine the value of w.You must indicate in your solution:1. The simplified differential equation in terms of the displacement x you will be solving2. The m equation and complimentary solution3. The choice for the particular solution and the actual particular solution xp4. Express the solution x as a piecewise…
Q3: Consider evaluation of different temperatures of solar photovoltaic/thermal system (PVT) as shown in Figure 1(a). The following set of differential equations represent energy balance equations to be solve using matrices and eigenvalues using MATLAB: dTglass = -0.75Tgtass + 0.75TpvT (1) dt - 1.187glass – 22Tpyr + 23Twax (2) dt dTwax 12Tglass + 18TpyT – 19 Twax (3) dt Where, Tgtass , TPVT, and Twax, are temperatures illustrated in Figure 1(b). At time t-0 the initial conditions are Tglass = 35 , Tpyr = 33, and Twax = 31 °C. Cold suppty In frem water Tank Glass PVT Espann Nane-PCMPVT Collector Wax Tubes Sterg Tank Nanofluid Heat Exchanger Contalner Tepe Teek oe Pump for drain