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To determine the effect of the temperature dependence of the thermal conductivity on the temperature distribution in a solid, consider a material for which this dependence may be represented as
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Fundamentals of Heat and Mass Transfer
- To determine the thermal conductivity of a structural material, a large 15-cm-thick slab of the material is subjected to a uniform heat flux of 2500 W/m2 while thermocouples embedded in the wall at 2.5 cm. intervals are read over a period of time. After the system had reached equilibrium, an operator recorded the thermocouple readings shown below for two different environmental conditions: Distance from the Surface (cm) Temperature (C) Test 1 0 40 5 65 10 97 15 132 Test 2 0 95 5 130 10 168 15 208 From these data, determine an approximate expression for the thermal conductivity as a function of temperature between 40 and 208C.arrow_forward1.63 Liquid oxygen (LOX) for the space shuttle is stored at 90 K prior to launch in a spherical container 4 m in diameter. To reduce the loss of oxygen, the sphere is insulated with superinsulation developed at the U.S. National Institute of Standards and Technology's Cryogenic Division; the superinsulation has an effective thermal conductivity of 0.00012 W/m K. If the outside temperature is on the average and the LOX has a heat of vaporization of 213 J/g, calculate the thickness of insulation required to keep the LOX evaporation rate below 200 g/h.arrow_forwardA plane wall 15 cm thick has a thermal conductivity given by the relation k=2.0+0.0005T[W/mK] where T is in kelvin. If one surface of this wall is maintained at 150C and the other at 50C, determine the rate of heat transfer per square meter. Sketch the temperature distribution through the wall.arrow_forward
- 1.3 A furnace wall is to be constructed of brick having standard dimensions of Two kinds of material are available. One has a maximum usable temperature of 1040°C and a thermal conductivity of 1.7 W/(m K), and the other has a maximum temperature limit of 870°C and a thermal conductivity of 0.85 W/(m K). The bricks have the same cost and are laid in any manner, but we wish to design the most economical wall for a furnace with a temperature of 1040°C on the hot side and 200°C on the cold side. If the maximum amount of heat transfer permissible is 950 , determine the most economical arrangement using the available bricks.arrow_forward2.15 Suppose that a pipe carrying a hot fluid with an external temperature of and outer radius is to be insulated with an insulation material of thermal conductivity k and outer radius . Show that if the convection heat transfer coefficient on the outside of the insulation is and the environmental temperature is , the addition of insulation actually increases the rate of heat loss if , and the maximum heat loss occurs when . This radius, is often called the critical radius.arrow_forward1.77 Explain each in your own words. (a) What is the mode of heat transfer through a large steel plate that has its surfaces at specified temperatures? (b) What are the modes when the temperature on one surface of the steel plate is not specified, but the surface is exposed to a fluid at a specified temperature?arrow_forward
- Derive a formula for the thermal resistance, R₁, for a spherical shell assuming one- dimensional heat flux in the radial direction. The inside and outside radii of the spherical shell are r; and r., respectively, and the shell is made of a material having thermal conductivity k. Assume the temperatures at the inside and outside surfaces of the shell are T and T., respectively. The thermal resistance formula assumes the rate of heat transfer through the spherical shell, Q, is constant. The heat flux is in the radial direction for this one-dimensional case, and the dT heat flux is given by Fourier's law: q, = -k The rate of heat transfer through a dr spherical surface is Q =q₁A where A = 4лr². Derive the formula for the thermal 1 ++)) r = resistance for a spherical shell answer: R₁ 1 1 4лk riarrow_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_forwardThe temperature distribution in a certain plane wall is: T-Ty =C₁+Cx²+C₁x² Where TI and T2 are the temperatures on each side of the wall. If the thermal conductivity of the wall is constant and the wall thickness is L, derive an expression for the heat generation per unit volume as a function of x, the distance from the plane where T-T1. Let the heat- generation rate be 'q0 at x = 0arrow_forward
- Question # 3: Design a heat equation model to determine the solution. The physical data needed to model for copper includes: density p= 5.82g/cm', thermal conductivity K = 0.095cal / cm sec C, %3D specific heat O =0.097cal / g C. The initial temperature is 70 sin(T x/100) C and the ends %3D are kept at 0 C of a laterally insulated copper bar of 120 cm long. Determine the time its goint to take for the maximum temperature in the bar to drop to 100 C ?arrow_forwardThe side of a home can be modeled as a plane wall. The wall is 12 centimeters thick. If the outdoor temperature is 30oC and the indoor temperature is 220C, what is the temperature in the wall 1 cm from the inside surface? Assume a steady-state temperature distribution. Draw a schematic or diagram. Show all calculations.arrow_forwardWhat will be the rise in temperature in 30 minutes of a block of copper of 500-gram mass if it is joined to a cylindrical copper rod 20 cm long and 3.0 mm in diameter when a temperature difference of 80 degree Celsius is maintained between ends of the rod? The thermal conductivity of copper is 1.02 cal/cm2-sec-°C/cm (neglect heat losses). Please include FBDarrow_forward
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning