Consider the triangular panel made of two different isotropic materials as shown in Fig. 2. A constant temperature of T = 0°C is prescribed along edge 2–3. Edge 1–2 is fully insulated. A linearly-varying heat flux of q = 15x W m-1 is applied along edge 1-3. Heat sources can be ignored. The panel is approximated using two linear triangular elements as indicated in Fig. 2. 2 (1) (1) q = 0 T = 0°C 3 m N(1) N(2) k = 4W°C-1 |k = 8 W °C-1 (2) (3) (2) 3 -1 q = 15x W m' 2 m 2 m Figure 2: Bi-material triangular panel. (a) Compute the element conductance matrix Ke for element 1. (b) Compute the global flux vector f. (c) Compute the global conductance matrix after imposition of the essential bound- ary conditions.

Consider the triangular panel made of two different isotropic materials as shown in Fig. 2. A constant temperature of T = 0°C is prescribed along edge 2–3. Edge 1–2 is fully insulated. A linearly-varying heat flux of q = 15x W m-1 is applied along edge 1-3. Heat sources can be ignored. The panel is approximated using two linear triangular elements as indicated in Fig. 2. 2 (1) (1) q = 0 T = 0°C 3 m N(1) N(2) k = 4W°C-1 |k = 8 W °C-1 (2) (3) (2) 3 -1 q = 15x W m' 2 m 2 m Figure 2: Bi-material triangular panel. (a) Compute the element conductance matrix Ke for element 1. (b) Compute the global flux vector f. (c) Compute the global conductance matrix after imposition of the essential bound- ary conditions.

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.

Chapter2: Steady Heat Conduction

Section: Chapter Questions

Problem 2.15P: 2.15 Suppose that a pipe carrying a hot fluid with an external temperature of and outer radius is...

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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.
As a designer working for a major electric appliance manufacturer, you are required to estimate the amount of fiberglass insulation packing (k = 0.035 W/m K) that is needed for a kitchen oven shown in the figure below. The fiberglass layer is to be sandwiched between a 2-mm-thick aluminum cladding plate on the outside and a 5-mm-thick stainless steel plate on the inside that forms the core of the oven. The insulation thickness is such that the outside cladding temperature does not exceed 40C when the temperature at the inside surface of the oven is 300C. Also, the air temperature in the kitchen varies from 15Cto33C, and the average heat transfer coefficient between the outer surface of the oven and air is estimated to be 12.0W/m2K. Determine the thickness of the fiberglass insulation that is required for these conditions. What would be the outer surface temperature when the inside surface of the oven is at 475C?
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.
1.2 The weight of the insulation in a spacecraft may be more important than the space required. Show analytically that the lightest insulation for a plane wall with a specified thermal resistance is the insulation that has the smallest product of density times thermal conductivity.
2.30 An electrical heater capable of generating 10,000 W is to be designed. The heating element is to be a stainless steel wire having an electrical resistivity of ohm-centimeter. The operating temperature of the stainless steel is to be no more than 1260°C. The heat transfer coefficient at the outer surface is expected to be no less than in a medium whose maximum temperature is 93°C. A transformer capable of delivering current at 9 and 12 V is available. Determine a suitable size for the wire, the current required, and discuss what effect a reduction in the heat transfer coefficient would have. (Hint: Demonstrate first that the temperature drop between the center and the surface of the wire is independent of the wire diameter, and determine its value.)
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?
1.37 Mild steel nails were driven through a solid wood wall consisting of two layers, each 2.5-cm thick, for reinforcement. If the total cross-sectional area of the nails is 0.5% of the wall area, determine the unit thermal conductance of the composite wall and the percent of the total heat flow that passes through the nails when the temperature difference across the wall is 25°C. Neglect contact resistance between the wood layers.
A 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.
2.50 Compare the rate of heat flow from the bottom to the top of the aluminum structure shown in the sketch below with the rate of heat flow through a solid slab. The top is at v, the bottom at . The holes are filled with insulation that does not conduct heat appreciably.
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.
Wearing layers of clothing in cold weather is often recommended because dead-air spaces between the layers keep the body warm. The explanation for this is that the heat loss from the body is less. Compare the rate of heat loss for a single 2-cm-thick layer of wool [k=0.04W/(mK)] with three 0.67-cm layers separated by 1.5 mm air gaps. The thermal conductivity of air is 0.024 W(mK).
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 ?