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57 Pages

### sm2_35

Course: MAE Heat Trans, Spring 2010
School: Seoul National
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Word Count: 338

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2.35 PROBLEM KNOWN: Three-dimensional system described by cylindrical coordinates (r,,z) experiences transient conduction and internal heat generation. FIND: Heat diffusion equation. SCHEMATIC: See also Fig. 2.9. ASSUMPTIONS: (1) Homogeneous medium. ANALYSIS: Consider the differential control volume identified above having a volume given as V = drrddz. From the conservation of energy requirement, q r q r +dr +...

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2.35 PROBLEM KNOWN: Three-dimensional system described by cylindrical coordinates (r,,z) experiences transient conduction and internal heat generation. FIND: Heat diffusion equation. SCHEMATIC: See also Fig. 2.9. ASSUMPTIONS: (1) Homogeneous medium. ANALYSIS: Consider the differential control volume identified above having a volume given as V = drrddz. From the conservation of energy requirement, q r q r +dr + q q +d + q z q z+dz + E g = E st . The generation and storage terms, both representing volumetric phenomena, are E g = qV = q ( dr rd dz ) Eg = Vc T/ t = ( dr rd dz ) c T/ t. (2,3) (1) Using a Taylor series expansion, we can write q r+dr = q r + q d , ( q r ) dr, q +d = q + r () q z+dz = q z + ( q z ) dz. z (4,5,6) Using Fouriers law, the expressions for the conduction heat rates are q r = kA r T/ r = k ( rd dz ) T/ r q = kA T/r = k ( dr dz ) T/r (7) (8) (9) q z = kA z T/ z = k ( dr rd ) T/ z. Note from the above, right that schematic the gradient in the -direction is T/r and not T/. Substituting Eqs. (2), (3) and (4), (5), (6) into Eq. (1), T ( q r ) dr q d ( q z ) dz + q dr rd dz = ( dr rd dz ) c . r z t () (10) Substituting Eqs. (7), (8) and (9) for the conduction rates, find T k ( rd dz ) r dr r T T k ( drdz ) r d z k ( dr rd ) z dz (11) T . t Dividing Eq. (11) by the volume of the CV, Eq. 2.24 is obtained. + q dr rd dz = ( dr rd dz ) c 1 T 1 T T T kr r + 2 k + z k z + q = c t r r r < Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.
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Seoul National - MAE - Heat Trans
PROBLEM 2.36KNOWN: Three-dimensional system described by spherical coordinates (r,) experiences transient conduction and internal heat generation. FIND: Heat diffusion equation. SCHEMATIC: See Figure 2.13. ASSUMPTIONS: (1) Homogeneous medium. ANALYSIS: T
Seoul National - MAE - Heat Trans
PROBLEM 2.37KNOWN: Temperature distribution in steam pipe insulation. FIND: Whether conditions are steady-state or transient. Manner in which heat flux and heat rate vary with radius. SCHEMATIC:ASSUMPTIONS: (1) One-dimensional conduction in r, (2) Const
Seoul National - MAE - Heat Trans
PROBLEM 2.38KNOWN: Inner and outer radii and surface temperatures of a long circular tube with internal energy generation. FIND: Conditions for which a linear radial temperature distribution may be maintained. SCHEMATIC:ASSUMPTIONS: (1) One-dimensional,
Seoul National - MAE - Heat Trans
PROBLEM 2.39KNOWN: Radii and thermal conductivity of conducting rod and cladding material. Volumetric rate of thermal energy generation in the rod. Convection conditions at outer surface. FIND: Heat equations and boundary conditions for rod and cladding.
Seoul National - MAE - Heat Trans
PROBLEM 2.40KNOWN: Steady-state temperature distribution for hollow cylindrical solid with volumetric heat generation. FIND: (a) Determine the inner radius of the cylinder, ri, (b) Obtain an expression for the volumetric rate of heat generation, q, (c) D
Seoul National - MAE - Heat Trans
PROBLEM 2.40 (Cont.)q ro, z = 16 W / m K 2 150C / m 1 m 12C /1 m = 4608 W / m r()22&lt; &lt;q r ( ro ) = A r q ro, z r()2whereA r = 2 ro ( 2z o )q r ( ro ) = 4 1 m 2.5 m 4608 W / m = 144, 765 WNote that the sign of the heat flux and heat rate in the
Seoul National - MAE - Heat Trans
PROBLEM 2.42KNOWN: Temperature distribution in a spherical shell. FIND: Whether conditions are steady-state or transient. Manner in which heat flux and heat rate vary with radius. SCHEMATIC:ASSUMPTIONS: (1) One-dimensional conduction in r, (2) Constant
Seoul National - MAE - Heat Trans
PROBLEM 2.43 KNOWN: Spherical container with an exothermic reaction enclosed by an insulating material whose outer surface experiences convection with adjoining air and radiation exchange with large surroundings. FIND: (a) Verify that the prescribed tempe
Seoul National - MAE - Heat Trans
PROBLEM 2.44 KNOWN: One-dimensional system, initially at a uniform temperature Ti, is suddenly exposed to a uniform heat flux at one boundary, while the other boundary is insulated. FIND: (a) Proper form of heat equation and boundary and initial condition
Seoul National - MAE - Heat Trans
PROBLEM 2.45 KNOWN: Plate of thickness 2L, initially at a uniform temperature of Ti = 200C, is suddenly quenched in a liquid bath of T = 20C with a convection coefficient of 100 W/m K. FIND: (a) On T-x coordinates, sketch the temperature distributions for
Seoul National - MAE - Heat Trans
PROBLEM 2.46KNOWN: Plane wall, initially at a uniform temperature, is suddenly exposed to convective heating. FIND: (a) Differential equation and initial and boundary conditions which may be used to find the temperature distribution, T(x,t); (b) Sketch T
Seoul National - MAE - Heat Trans
PROBLEM 2.47KNOWN: Plane wall, initially at a uniform temperature Ti, is suddenly exposed to convection with a fluid at T at one surface, while the other surface is exposed to a constant heat flux q . o FIND: (a) Temperature distributions, T(x,t), for in
Seoul National - MAE - Heat Trans
PROBLEM 2.48KNOWN: Plane wall, initially at a uniform temperature To, has one surface (x = L) suddenly exposed to a convection process (T &gt; To,h), while the other surface (x = 0) is maintained at To. Also, wall experiences uniform volumetric heating q su
Seoul National - MAE - Heat Trans
PROBLEM 2.49KNOWN: Plane wall, initially at a uniform temperature To, has one surface (x = L) suddenly exposed to a convection process (T &lt; To, h), while the other surface (x = 0) is maintained at To. Also, wall experiences uniform volumetric heating q s
Seoul National - MAE - Heat Trans
PROBLEM 2.50KNOWN: Size and thermal conductivities of a spherical particle encased by a spherical shell. FIND: (a) Relationship between dT/dr and r for 0 r r1, (b) Relationship between dT/dr and r for r1 r r2, (c) Sketch of T(r) over the range 0 r r2. SC
Seoul National - MAE - Heat Trans
PROBLEM 2.51KNOWN: Temperature distribution in a plane wall of thickness L experiencing uniform volumetric heating q having one surface (x = 0) insulated and the other exposed to a convection process characterized by T and h. Suddenly the volumetric heat
Seoul National - MAE - Heat Trans
PROBLEM 3.2KNOWN: Temperatures and convection coefficients associated with air at the inner and outer surfaces of a rear window. FIND: (a) Inner and outer window surface temperatures, Ts,i and Ts,o, and (b) Ts,i and Ts,o as a function of the outside air
Seoul National - MAE - Heat Trans
PROBLEM 3.3KNOWN: Desired inner surface temperature of rear window with prescribed inside and outside air conditions. FIND: (a) Heater power per unit area required to maintain the desired temperature, and (b) Compute and plot the electrical power require
Seoul National - MAE - Heat Trans
PROBLEM 3.4KNOWN: Curing of a transparent film by radiant heating with substrate and film surface subjected to known thermal conditions. FIND: (a) Thermal circuit for this situation, (b) Radiant heat flux, q (W/m2), to maintain bond at o as a function of
Seoul National - MAE - Heat Trans
PROBLEM 3.7 KNOWN: A layer of fatty tissue with fixed inside temperature can experience different outside convection conditions. FIND: (a) Ratio of heat loss for different convection conditions, (b) Outer surface temperature for different convection condi
Seoul National - MAE - Heat Trans
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