Concept explainers
Consider a medium in which the heat conduction equation is given in its simplest forms as
Is heat transfer steady or transient?
Is heat transfer one-. two-, or thee-dimensional?
Is there heat generation in the medium?
Is the thermal conductivity of the medium constant or variable?
Is the medium a plane wall, a cylinder, or a sphere?
Is this differential equation for heat conduction linear or nonlinear?
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Heat and Mass Transfer: Fundamentals and Applications
- 2.34 Show that the temperature distribution in a sphere of radius . made of a homogeneous material in which energy is released at a uniform rate per unit volume , isarrow_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_forwardShow that the rate of heat conduction per unit length through a long, hollow cylinder of inner radius ri and outer radius ro, made of a material whose thermal conductivity varies linearly with temperature, is given by qkL=TiTo(rori)/kmA where Ti = temperature at the inner surface To = temperature at the outer surface A=2(rori)/ln(ro/ri)km=ko[1+k(Ti+To)/2]L=lenthofcyclinderarrow_forward
- 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_forward2.38 The addition of aluminum fins has been suggested to increase the rate of heat dissipation from one side of an electronic device 1 m wide and 1 m tall. The fins are to be rectangular in cross section, 2.5 cm long and 0.25 cm thick, as shown in the figure. There are to be 100 fins per meter. The convection heat transfer coefficient, both for the wall and the fins, is estimated to be K. With this information determine the percent increase in the rate of heat transfer of the finned wall compared to the bare wall.arrow_forward
- The temperature distribution across a wall 0.3 m thick at a certain instant of time is T(x) = a+ b+cx?, where T is in degrees Celsius and x is in meters, a = 200°C,b = -200°, and c = conductivity of 1 W /m · K. 30°C/m² . The wall has a thermal (a) On a unit surface area basis, determine the rate of heat transfer into and out of the wall and the rate of change of energy stored by the wall. (b) If the cold surface is exposed to a fluid at 100°C, what is the convection coefficient? k=1W/m•k T(x) =200-200x + 30x² 200°C- ĖST 142.7°C q"out | Fluid Too = 100°C,h 9"in |L-0.3marrow_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_forwardQ1: In a nuclear reactor, heat is generated uniformly in the 5-cm-diameter cylindrical uranium rods at a rate of 7 x 10' W/m. If the length of the rods is 1 m, determine the rate of heat generation in each rod. Answer: Q=137.4 kW Q2: Write down the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation in its simplest form, and indicate what each term represents. Also, what is the units of every term? Q3: Consider a spherical container of inner radius r1, outer radius r2, and thermal conductivity k. Express the boundary condition on the inner surface of the container for steady one dimensional conduction for the following cases: (a) specified temperature of 50°C, (b) specified heat flux of 30 W/m² toward the center, (c) convection to a medium at T with a heat transfer coefficient of h. Q4: Heat is generated in a long wire of radius ro at a constant rate of go per unit volume. The wire is covered with a plastic…arrow_forwardThe rate of heat conduction through a plane wall is proportional to the average thermal conductivity, the wall area, and the temperature difference, but is inversely proportional to the ---------------.arrow_forward1- A solid infinitely long cylinder, radius 2 cm, has uniform internal heat generation. The temperature distribution in the cylinder is T(r) = = 256 – 8.6 x 104 r² where r is in meters, T in °C and the thermal conductivity of the cylinder material is 16 W/ m °C. Determine: (a) The temperature at the centerline. (b) The surface temperature. (c) The heat flux at the surface. (d) The rate of heat transfer to the surrounding per unit meter of cylinder length.arrow_forwardAnswer step by step. Subject: thermodynamics and Heat Transferarrow_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