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Derive an expression for the temperature distribution in a plane wall in which there are uniformly distributed heat sources that vary according to the linear relation
where
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Chapter 2 Solutions
Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
- A plane wall, 7.5 cm thick, generates heat internally at the rate of 105 W/m3. One side of the wall is insulated, and the other side is exposed to an environment at 90C. The convection heat transfer coefficient between the wall and the environment is 500 W/m2 K. If the thermal conductivity of the wall is 12 W/m K, calculate the maximum temperature in the wall.arrow_forwardDiscuss the modes of heat transfer that determine the equilibrium temperature of the space shuttle Endeavour when it is in orbit. What happens when it reenters the earths atmosphere?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_forward
- 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_forwardA plane wall of thickness 2L = 2*33 mm and thermal conductivity k = 7 W/m-K experiences uniform volumetric heat generation at a rate q˙, while convection heat transfer occurs at both of its surfaces (x = −L, + L), each of which is exposed to a fluid of temperature T∞ = 31°C. Under steady-state conditions, the temperature distribution in the wall is of the form T(x) = a + bx + cx2 where a = 85°C, b = −-218°C/m, c = −-23,942°C/m2, and x is in meters. The origin of the x-coordinate is at the midplane of the wall. (a) Sketch the temperature distribution and identify significant physical features. (b) What is the volumetric rate of heat generation q˙ in the wall? (c) Obtain an expression for the heat flux distribution qx″(x). Is the heat flux zero at any location? Explain any significant features of the distribution. (d) Determine the surface heat fluxes, qx″(−L) and qx″(+L). How are these fluxes related to the heat generation rate? (e) What are the convection coefficients…arrow_forward1-D, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/m/K. The temperature distribution has the form T = a + bx + cx² °C. The surface at x=0 has a temperature of To = 120 °C and experiences convection with a fluid for which T.. surface at x= 50 mm is well insulated (no heat transfer). Find: (a) The volumetric energy generation rate q. (15) (b) Determine the coefficients a, b, and c. 20 °C and h 500 W/m² K. The To: = 120°C T = 20°C h = 500 W/m².K 111 Fluid T(x)- = q, k = 5 W/m.K L = 50 mmarrow_forward
- Steady state temperatures at three nodes are given in K. This object generates heat itself at rate of q = 5×107 W/m³ and has a thermal conductivity of 20 W/m K. Two of its sides are maintained at a constant temperature of 300 K, while the others are insulated. Find temperatures at nodes 1, 2 and 3 in K. 5 mm 2 398.0 348.5 3 374.6 - Uniform temperature, 300 K 5 mmarrow_forward3.10 By neglecting lateral temperature variation in the analysis of fins, h,T. 木 H two-dimensional conduction is modeled as a one-dimensional H problem. То examine this T, h,T. approximation, consider a semi- infinite plate of thickness 2H. The base is maintained at uniform temperature T,. The plate exchanges heat by convection at its semi- infinite surfaces. The heat transfer coefficient is h and the ambient temperature is T.. Determine the heat transfer rate at the base.arrow_forwardConsider a heat conductor in the form of a long cylinder, with inner and outer radii R1 and R2, respectively. Heat is generated within the cylinder, where the temperature O(r, t) at position r and time t satisfies the modified heat equation = DV0 + H, where D is the thermal diffusivity, and H is proportional to the rate of heat production. The inner and outer surfaces of the cylinder are cooled by a fluid maintained at constant temperature Oo. (a) If the temperature is in a steady state and depends only on the distance r from the centre of the cylinder, use cylindrical coordinates (r, 0, 2) to write down an ordinary differential equation for O(r) valid in the region R1arrow_forward= Consider a large plane wall of thickness L=0.3 m, thermal conductivity k = 2.5 W/m.K, and surface area A = 12 m². The left side of the wall at x=0 is subjected to a net heat flux of ɖo = 700 W/m² while the temperature at that surface is measured to be T₁ = 80°C. Assuming constant thermal conductivity and no heat generation in the wall, (a) express the differential equation and the boundary equations for steady one- dimensional heat conduction through the wall, (b) obtain a relation for the variation of the temperature in the wall by solving the differential equation, and (c) evaluate the temperature of the right surface of the wall at x=L. Ti до L Xarrow_forwardA new 1 ft thick insulating material was recently tested for heat resistant properties. The data recorded temperatures of 70 deg. F and 210 deg. F on the cold and hot sides, respectively. If the thermal conductivity of the insulating material is 0.026 Btu/ft . h .⁰ F, calculate the rate of the heat flux,Q/A, through the wall in Btu/ft^2 . h. Resolve the problem in SI units.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 ρ =1200 kg/m 3, k=400 W/m.K, and cp=2500 J/kg.K, respectively. A uniform heat flux q =1000 W/m 2 is applied to the upper surface. The right and left surfaces are also kept at 0oC. Bottom surface is insulated.arrow_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