Find the temperature distribution in a rod (Fig. P31.11) with internal heat generation using the finite-element method. Derive the element nodal equations using Fourier heat conduction
and heat conservation relationships
where
Develop the nodal equations that must be solved for the temperatures and temperature gradients at each of the six nodes. Assemble the equations, insert the boundary conditions, and solve the resulting set for the unknowns.
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Chapter 31 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
- In the Fig. 2 below, let Ki = K2 = K and ti = t=t. %3D T -T X Fig. 2 (a) Let T= 0 °C and T= 200 °C. Solve for T: and unknown rates of heat flow in term of k and t. MEC_AMO_TEM_035_02 Page 2 of 11 Finite Element Analysis (MECH 0016.1) – Spring - 2021 -Assignment 2-QP (b) Let T- 400 °C and let fs have the prescribed value f. What are the unknowns? Solve for them in term of K, t, and f.arrow_forwardThe 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_forwardThe steady-state temperature distribution in a one-dimensional wall of thermal conductivity k=83 (W/m) °C) and thickness 300 mm is observed to be T(°C) = ax2+bx+c, where a = -2500 °C/m2 , b = 500 °C/m, c = 250 °C and x is in meters. a) What is the heat generation rate ?̇ (W/m3 ) in the wall ? b) Find the maximum and minimum temperatures in the wall. c) Find the amount of heat transferred to the right side of the wall (for 1 m2 surface area.)arrow_forward
- A food product containing a water content of 85% is frozen. Estimate the heat type products at -10 °C when 82% of the water is in its frozen state. Hot product type dry is 2.5 kJ/(kg °C) is assumed to heat this type of water at -10 °C is equal to heat water type at 0 °C, and the heat of the types of ice follow function Cp ice = 0 0062 x T frozen + 2 0649. a. Cp frozen products kJ/kg °C.arrow_forwardi. Derive the linear and quatratic approximation of the below resistance temperature readings(temperature from 40 °c to 80 °c). (. ii. find out the resistance of RTD at a temperature of 200°c using linear approximation? ( Temperature 40 45 50 55 60[TO] 65 70 75 80 Resistance ohm 115.239 120.521 167.096 254.966 384.13 554.587 766.34 1019.38 1313.724arrow_forwardI.C 02/A/ Use the Crank-Nicolson method to solve for the temperature distribution of a long thin rod with a length of 10 cm and the following values: k = 0.49 cal/(s cm °C), Ax = 2 cm, and At = st 0.1 s. Initially the temperature of the rod is 0°C and the boundary conditions are fixed for all times at 7(0, t) = 100°C and 7(10, t) = 50°C. Note that the rod is aluminum with C = 0.2174 cal/g °C) and p = 2.7 g/cm³. List the tridiagonal system of equations and determined the temperature up to 0.1 s.arrow_forward
- a) Separate the rod into 4 control sections, each with a node in the centre, then use finite-volume analysis to estimate the temperature along the length of the rod. b) Separate the rod into 4 sections and repeat (a) with equal both temperature values and beginning conditions using the finite difference method (if necessary). c) Write down and analytically compare with (a) and (b) differential equation for the temperature distribution along the rod. d) Compare the results (a), (b) and (c) by using a graph and differences.e) How to improve the results from (a) and (b)? Show it.arrow_forwardA// Use Implicit Method to solve the temperature distribution of a long thin rod with a length of 9 cm and following values: k = 0.49 cal/(s cm °C), Ax = 3 cm, and At = 0.2 s. At t=0 s, the temperature of the rod is 10°C and the boundary conditions are fixed dT (9,t) 1 °C/cm. Note that the rod for alltimes at 7(0,t) = 80°C and derivative condition dx is aluminum with C = 0.2174 cal/g °C) and p = 2.7 g/cm³. Find the temperature values on the inner grid points and the right boundary for t = 0.4 s.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_forward
- QUESTION-) 236 °C steam flows in a pipe which features are given below. Inner and outer diameters of pipe are 300 mm and 320 mm. Coefficient of heat transmission is 40 W/mK. It's external ambient is air and it's temperature is 20 °C. Take the temperature of the pipe's external surface 180 °C. It is known that the internal temp convection coefficient is 600 W/m^2K and external heat transfer coefficient is 5,9109 W/m^2K The pipe's length is 500 meter. You need to add the radiation in the calculation. The rate of the radiation emissivity is 0.85. You can take the environmental surface temp directly. Please calculate the heat loss of pipe.arrow_forwardQUESTION-) 236 °C steam flows in a pipe which features are given below. Inner and outer diameters of pipe are 300 mm and 320 mm. Coefficient of heat transmission is 40 W/mK. It's external ambient is air and it's temperature is 20 °C. Take the temperature of the pipe's external surface 180 °C. It is known that the internal temp convection coefficient is 600 W/m^2K and external heat transfer coefficient is 5,9109 W/m^2K The pipe's length is 500 meter. You need to add the radiation in the calculation. The rate of the radiation emissivity is 0.85. You can take the environmental surface temp directly. a-) Please draw this question on the resistance network. b-) Please calculate the heat loss of pipe. c-) By applying insulation to this pipe, it is desired to reduce the heat loss to 20% of the uninsulated loss (option a). Calculate the insulation thickness accordingly. Take glass wool as insulation material.(Conductivity of glass wool = 0.040 W/mK.) Given, The temperature of steam inside the…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
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning
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