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Conduction within relatively complex geometries can sometimes be evaluated using the finite-difference methods of this text that are applied to subdomains and patched together. Consider the two-dimensional domain formed by rectangular and cylindrical subdomains patched at the common, dashed control surface. Note that, along the dashed control surface, temperatures in the two subdomains are identical and local conduction heat fluxes to the cylindrical subdomain are identical to local conduction heat fluxes from the rectangular subdomain.
Calculate the heat transfer per unit depth into the page,
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Fundamentals of Heat and Mass Transfer
- Consider the square channel shown in the sketch operating under steady state condition. The inner surface of the channel is at a uniform temperature of 600 K and the outer surface is at a uniform temperature of 300 K. From a symmetrical elemental of the channel, a two-dimensional grid has been constructed as in the right figure below. The points are spaced by equal distance. Tout = 300 K k = 1 W/m-K T = 600 K (a) The heat transfer from inside to outside is only by conduction across the channel wall. Beginning with properly defined control volumes, derive the finite difference equations for locations 123. You can also use (n, m) to represent row and column. For example, location Dis (3, 3), location is (3,1), and location 3 is (3,5). (hint: I have already put a control volume around this locations with dashed boarder.) (b) Please use excel to construct the tables of temperatures and finite difference. Solve for the temperatures of each locations. Print out the tables in the spread…arrow_forward3. A thin metallic wire of thermal conductivity k, diameter D, and length 2L is annealed by passing an electrical current through the wire to induce a uniform volumetric heat generation åg. The ambient air around the wire is at a temperature To, while the ends of the wire at xarrow_forward25. Develop an algorithm, along with the program (in python), to find the temperature distribution in the Problem 5.102 NOTE: Use the explicit finite differences method 5.102 Consider the fuel element of Example 5.9. Initially, the element is at a uniform temperature of 250°C with no heat generation. Suddenly, the element is inserted into the reactor core causing a uniform volumetric heat generation rate of q = 108 W/m³. The surfaces are convectively cooled with T = 250°C and_h= 1100 W/m² K. Using the explicit method with a space increment of 2 mm, determine the temperature distribution 1.5 s after the element is inserted into the core. EXAMPLE 5.9 A fuel element of a nuclear reactor is in the shape of a plane wall of thickness 2L = 20 mm and is convectively cooled at both surfaces, with h = 1100 W/m². K and T=250°C. At normal operating power, heat is generated uniformly within the element at a volumetric rate of q₁ = 107 W/m³. A departure from the steady-state conditions associated…arrow_forwardFind the steady temperature distribution in the semi infinite plate shown below. The 2D steady heat conduction equation is: use the method of separation of variables.arrow_forward(a) Consider nodal configuration shown below. (a) Derive the finite-difference equations under steady-state conditions if the boundary is insulated. (b) Find the value of Tm,n if you know that Tm, n+1= 12 °C, Tm, n-1 = 8 °C, Tm-1, n = 10 °C, Ax = Ay = 10 mm, and k = = W 3 m. k . Ay m-1, n m, n | Δx=" m, n+1 m, n-1 The side insulatedarrow_forwardFig. 4 illustrates an insulating wall of three homogeneous layers with conductivities k1, k2, and k3 in intimate contact. Under steady state conditions, both right and left surfaces are exposed to a temperature in a steady state condition at ambient temperatures of T and T , respectively, while ß, and BLare the film coefficients respectively. Assume that there is no internal heat generation and that the heat flow is one-dimensional (dT/dy = 0). For the illustrated ambient temperature in Fig. 4, determine the temperature's distribution at each layer. Material 3 Material 1 Material 2 T= 100 T= 35 °C Kı=20 K3=50 (W/m.k) K3=30 (W/m.k) B1= 10 w/m² °K (W/m.k) BR= 15 w/m²°K 50 mm 35 mm 25 cm Fig. 4arrow_forwardWe have used linear one-dimensional elements to approximate the temperature distribution along a fin. The nodal temperatures and their corresponding positions are shown in Fig. 4. What is the temperature of the fin at (a) X= 4cm ( and (b) X 8cm MEC_AMO_TEM_035_02 Page 3 of 11 Finite Element Analysis (MECH 0016.1) - Spring - 2021 -Assignment 2-QP Td18°C T - S0°C ! (1) 2 (2) (3) °C 34 |T, T. 20 |-2 cm---3 cm- -Sem- Fig. 4arrow_forwardOne-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_forwardConsider a round potato being baked in an oven. Would you model the heat transfer to the potato as one-, two-, or three-dimensional by writing of the differential equations? (Steady state and no heat generation) Would you model the heat transfer for steady or transient system consisting of heat generation by writing of the differential equations? If the system is transient and consisting no heat generation, write initial boundary condition for one-dimensional the differential equation for the potato?arrow_forward(a) Consider nodal configuration shown below. (a) Derive the finite-difference equations under steady-state conditions if the boundary is insulated. (b) Find the value of Tm,n if you know that Tm, n+1= 12 °C, Tm, n-1 = 8 °C, Tm-1, n = 10 °C, Ax = Ay = 10 mm, and k = W 3 m. k Ay m-1, n 11- m2, 11 m, n+1 m, n-1 The side insulatedarrow_forward2. Consider the temperature distributions associated with a dx differential control volume within the one-dimensional plane walls shown below. T(x,00) T\x,00) * dx * dx (a) (Б) Tx,1) T(x,1) * dx dx (c) (d) (a) Steady-state conditions exist. Is thermal energy being generated within the differential control volume? If so, is the generation rate positive or negative? (b) Steady-state conditions exist as in part (a). Is the volumetric generation rate positive or negative within the differential control volume? (c) Steady-state conditions do not exist, and there is no volumetric thermal energy generation. Is the temperature of the material in the differential control volume increasing or decreasing with time? (d) Transient conditions exist as in part (c). Is the temperature increasing or decreasing with time?arrow_forwardDrive an expression for heat transfer and temperature distribution for steady state one dimensional heat conduction in a plan wall. The temperature is maintained at a temperature Ti at x=0, while the other face X-L is maintained at temperature T2, the thickness of the wall may be taken as L and the energy equation is given by: d²T/dx² = 0. : Sketch a simple diagram for the temperature distribution in plane wall for a steady state one dimensional heat conduction, with heat generation. The surface temperature of the walls Ti and T2, for the cases Ti>T2, T1-T2, and T2>T1. The thickness of the wall may be taken as 2Larrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning