1. Suppose you have a 4-foot metal rod, made out of steel which has a thermal conductive factor of a = 4 Assume the temperature on the ends of the rod is always 0 degrees. a) Filling in the numerical parameters mentioned above in the problem, write down the general solution to this heat equation, which should include some unknown coefficients (c,'s), in addition to t's and x's.

1. Suppose you have a 4-foot metal rod, made out of steel which has a thermal conductive factor of a = 4 Assume the temperature on the ends of the rod is always 0 degrees. a) Filling in the numerical parameters mentioned above in the problem, write down the general solution to this heat equation, which should include some unknown coefficients (c,'s), in addition to t's and x's.

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)

8th Edition

ISBN:9781305387102

Author:Kreith, Frank; Manglik, Raj M.

Publisher:Kreith, Frank; Manglik, Raj M.

Chapter1: Basic Modes Of Heat Transfer

Section: Chapter Questions

Problem 1.12P: 1.12 A wall with a thickness is made of a material with a thermal conductivity that varies with its...

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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.
2.15 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.
5.10 Experiments have been performed on the temperature distribution in a homogeneous long cylinder (0.1 m diameter, thermal conductivity of 0.2 W/m K) with uniform internal heat generation. By dimensional analysis, determine the relation between the steady-state temperature at the center of the cylinder , the diameter, the thermal conductivity, and the rate of heat generation. Take the temperature at the surface as your datum. What is the equation for the center temperature if the difference between center and surface temperature is when the heat generation is ?
A plane wall of thickness 2L has internal heat sources whose strength varies according to qG=qocos(ax) Where qo is the heat generated per unit volume at the center of the wall (x=0) and a is a constant. If both sides of the wall are maintained at a constant temperature of Tw, derive an expression for the total heat loss from the wall per unit surface area.
1.77 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?
1.63 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.
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.
1.12 A wall with a thickness is made of a material with a thermal conductivity that varies with its thickness according to the equation, , where and are constants. If the heat flux applied at the surface of one end of the wall is , derive an expression for the temperature gradient and the temperature distribution across the wall thickness (between and ). Use and define appropriate notations for the surface temperatures at each end of the wall.
For flow over a slightly curved isothermal surface, the temperature distribution inside the boundary layer t can be approximated by the polynomial T(y)=a+by+cy2+d3(yt), where y is the distance normal to the surface. (a) By applying appropriate boundary conditions, evaluate the constants a, b, c, and d. Fluid (b) Then obtain a dimensionless relation for the temperature distribution in the boundary layer.
A hollow sphere with inner and outer radii of R1 and R2, respectively, is covered with a layer of insulation having an outer radius of R3. Derive an expression for the rate of heat transfer through the insulated sphere in terms of the radii, the thermal conductivities, the heat transfer coefficients, and the temperatures of the interior and the surrounding medium of the sphere.
The 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.
You are tasked to design a cooling system for an ice rink. A standard ice rink has surface area ofArink = 1580 m2 . In this design, a technologically advanced solid state thermoelectric generatingcooling plate is placed in between concrete slabs. The following diagram contains the dimensionalparameters of the design (a) In the space below, with your best effort to correspond to the above diagram, draw a thermalcircuit that establishes the relationship between the cooling plate’s heat rate, Q, and the system’stemperatures and thermal resistances. Label the appropriate dimensions, thermal conductivities,convection coefficient, and temperatures. Ignore effects from contact resistance. (b) Given that the temperature at the top surface of the ice must be T ice = -5°C, obtain the requiredheat rate Q that must be drawn by the cooling plate in units Kilowatts. Be careful of +/- sign.Answer: ____________________________ [kW] c) Using the thermal circuit you established in Part (a), obtain the…