A two-dimensional rectangular plate is subjected to prescribedboundary conditions. Using the results of the exactsolution for the heat equation presented in Section 4.2,calculate the temperature at the midpoint
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Introduction to Heat Transfer
- A plane wall of thickness 8cm and thermal conductivity k=5W/mK experiences uniform volumetric heat generation, while convection heat transfer occurs at both of its surfaces (x= -L, x= + L), each of which is exposed to a fluid of temperature T∞ = 20˚C. The origin of the x-coordinate is at the midplane of the wall. Under steady-state conditions, the temperature distribution in the wall is of the form T(˚C) = a + bx - cx^2, where x is in meters, a =86˚C, b = -500˚C/m, and c=4459. 1) Heat Flux Entering the wall is ? 2) Temperature at the left face is /arrow_forwardFind the two-dimensional temperature distribution T(x,y) and midplane temperature T(B/2,W/2) under steady state condition. The density, conductivity and specific heat of the material are p=(1200*32)kg/mº, k=400 W/m.K, and cp=2500 J/kg.K, respectively. A uniform heat flux 9" =1000 W/m² is applied to the upper surface. The right and left surfaces are also kept at 0°C. Bottom surface is insulated. 9" (W/m) T=0°C T=0°C W=(10*32)cm B=(30*32)cmarrow_forward2. A slab of thickness Lis initially at zero temperature. For times t> 0, the boundary surface at x 0 is subjected to a time-dependent prescribed temperature f(t) defined by: a + bt for 0Ti and the boundary at x = L is kept insulated. Using Duhamel's theorem, develop an expression for the temperature distribution in the slab for times (i) t t1.arrow_forward
- Suppose that a finite-difference solution has been obtained for the temperature T, near but not at an adiabatic boundary. in most instances, it would be necessary or desirable to evaluate the temperature at the boundary point itself. for this case of an adiabatic boundary, develop an expression for the temperature at the boundary T1, in the terms of temperatures at neighbouring points T2, T3, etc, by assuming that the temperature distribution in the neighbourhood of the boundary is a straight line a second-degree polynomial a cubic polynomial (you only need to indicate how you would derive this one) indicate the order of T.E in each of the above approximations used to evaluate T1arrow_forward2. The slab shown is embedded in insulating materials on five sides, while the front face experiences convection off its face. Heat is generated inside the material by an exothermic reaction equal to 1.0 kW/m'. The thermal conductivity of the slab is 0.2 W/mk. a. Simplify the heat conduction equation and integrate the resulting ID steady form of to find the temperature distribution of the slab, T(x). b. Present the temperature of the front and back faces of the slab. n-20- 10 cm IT- 25°C) 100 cm 100 cmarrow_forwardThe initial temperature distribution of a 5 cm long stick is given by the following function. The circumference of the rod in question is completely insulated, but both ends are kept at a temperature of 0 °C. Obtain the heat conduction along the rod as a function of time and position ? (x = 1.752 cm²/s for the bar in question) 100 A) T(x1) = 1 Sin ().e(-1,752 (³¹)+(sin().e (-1,752 (²) ₁ + 1 3π TC3 .....) 100 t + ··· ....... 13) T(x,t) = 200 Sin ().e(-1,752 (²t) + (sin (3). e (-1,752 (7) ²) t B) 3/3 t + …............) C) T(x.t) = 200 Sin ().e(-1,752 (²t) (sin().e(-1,752 (7) ²) t – D) T(x,t) = 200 Sin ().e(-1,752 (²)-(sin().e (-1,752 (²7) ²) t E) T(x.t)=(Sin().e(-1,752 (²t)-(sin().e(-1,752 (²) t+ t + ··· .........) t +.... t + ··· .........) …..)arrow_forward
- A wall of a house is made from two layers of bricks enclosing a layer of insulation. A radiator is positioned to cover the whole internal surface, and used intermittently when the internal temperature is low. The external surface is exposed to the outside air. Which of the following assumptions could be used to identify the relevant reduced form of the conduction equation to find the temperature in the wall. a. Conduction is mainly in two directions. b. Conduction is mainly in one direction. c. The wall properties are homogeneous. d. Steady conditions exist. e. Unsteady conditions exist. f. There is an internal volumetric heat generation in the wall.arrow_forwardThe TPD method measures temperature elevations in a tissue region during a heating pulse and its later temperature decay after the pulse. It is then using the Pennes bioheat equation to perform a curve fitting to determine the local blood perfusion rate. If the TPD probe is placed in the vicinity of very large blood vessel, will the TPD technique provide an accurate measurement of the local blood perfusion in the vicinity of this large blood vessel? Explain briefly. (Hint: Is the Pennes bioheat equation accurate surrounding a large blood vessel?)arrow_forwardAn engineer seeks to study the effect of temperature on the curing of concrete by controlling the curing temperature in the following way. A sample slab of thickness L is subjected to a heat flux, qw, on one side, and it is cooled to temperature T1 on the other. Derive a dimensionless expression for the steady temperature in the slab. Plot the expression and offer a criterion for neglecting the internal heat generation in the slab.arrow_forward
- A two dimensional rectangular plate is subjected to prescribed boundary conditions. Using the results of the analytical solution for the heat equation presented in class, calculate the temperature at the midpoint (1,0.5) by considering the first five nonzero terms of the infinite series that must be evaluated. T₁ = 50°C y (m) 1 T₂ = 150°C T₁ = 50°C ►x (m) 2 -T₁ = 50°Carrow_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_forwardFind the steady-state temperature distribution in a metalic plate 20 cm by 60 cm if the two adjacent plates are held at 200 degree and the other two sides at zero degree.arrow_forward
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning