Consider the series solution, Equation 5.42, for the plane wall with convection. Calculate midplane
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Introduction to Heat Transfer
- Solve the heat equation u, = 0.2u on a digital computer using XX the Crank-Nicolson scheme. for the initial condition and boundary conditions TABLE P4.1 Case 1 2 3 4 5 Number of Grid Points 11 11 16 11 11 (x,0) = 100 sin TX L 0.25 0.50 0.50 1.00 2.00 L = 1 u(0,t) = u(L,t) = 0 Compute to t = 0.5 using the parameters in Table P4.1 (if possible) and compare graphically with the exact solution.arrow_forward1. A spring mass system serving as a shock absorber under a car's suspension, supports the M 1000 kg mass of the car. For this shock absorber, k = 1 × 10°N /m and c = 2 × 10° N s/m. The car drives over a corrugated road with force %3| F = 2× 10° sin(@t) N . Use your notes to model the second order differential equation suited to this application. Simplify the equation with the coefficient of x'" as one. Solve x (the general solution) in terms of w using the complimentary and particular solution method. In determining the coefficients of your particular solution, it will be required that you assume w – 1z w or 1 – o z -w. Do not use 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 solving 2. The m equation and complimentary solution xe 3. The choice for the particular solution and the actual particular solution x,…arrow_forwardOne of the strengths of numerical methods is their ability to handle complex boundary conditions. In the sketch, the boundary condition changes from specified heat flux ′′ qs (into the domain) to convection, at the location of the node (m, n). Write the steady-state, two- dimensional finite difference equation at this node.arrow_forward
- The steady-state distribution of temperature on a heated plate can be modeled by the Laplace equation, a²T ²T + a²x a²y If the plate is represented by a series of nodes as illustrated in Figure, centered finite-divided differences can be substituted for the second derivatives, which results in a system of linear algebraic equations. Use the Gauss-Seidel method to solve for the temperatures of the nodes in Figure. 0= Submission date: 09/01/2024 25°C T12 T₂2 250°C # T₁1 T₂1 250 CO 75°C 25°C 75°C 0°C 0°Carrow_forwardConsider the following linear equations,arrow_forwardLet's assume that the outdoor temperature in your region was 1 C on 26.12.2002. Let's assume that you use a 2088 W heater in the room in order to keep the indoor temperature of the room at 20 ° C. In the meantime, a 68 W light bulb for lighting, a computer you use to solve this question and load it into the system (let's assume it consumes 217 W of energy), you and your two friends (three people in total) are in the room to assist you in solving the questions. A person radiates 45 J of heat per second to his environment. When you consider all these conditions, calculate the exergy destruction caused by the heat loss from the exterior wall of your room.arrow_forward
- One-dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall which is subject to convection on the left side at x = 0 and being well-insulated on the other.a) Specify the mathematical model defining T(x): provide a governing differential equation and appropriate boundary conditions. Express your answer in terms of defined variables rather than numerical values with units. b) Solve for the temperature profile T(x) referencing the x-origin as shown on the left surface (again expressing your answer in terms of defined variables rather than numericalvalues.) c) Find the maximum temperature in the wall and the wall surface temperature if the volumetric generation is qdot = 1 MW/m^3 with the remaining parameters as specified in the figure.arrow_forwardQuestion 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 constantarrow_forwardDifferential 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.arrow_forward
- A two dimensional square plate (with 2m on each side) is subjected to the boundary conditions shown below. y T= 300 °C T= 70 °C 3 m T= 70°C 3 m T= 70 °C 1) Plot the temperature distribution obtained by the numerical solution a. using a uniform grid size of 0.5 m (Ax=Ay=0.5) b. using a uniform grid size of 0.1 m (Ax=Ay=0.1) 2) Plot temperatures (obtained by exact and two numerical solutions) as a function of a. x at y=1.0 m b. x at y=1.5 m c. y at x-1.0 m d. y at x-1.5 marrow_forwardCircle final answer Question: Which series helps us to derive the discretized form of the governing equations? Remember to circle final answerarrow_forwardQuestion 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 constantarrow_forward
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning