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2 Pages

### problem4-05

Course: MAE Heat Trans, Spring 2010
School: Seoul National
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Word Count: 224

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4.5 KNOWN: PROBLEM Boundary conditions on four sides of a rectangular plate. FIND: Temperature distribution. SCHEMATIC: y q s W T1 T1 0 0 T1 L x ASSUMPTIONS: (1) Two-dimensional, steady-state conduction, (2) Constant properties. ANALYSIS: This problem differs from the one solved in Section 4.2 only in the boundary condition at the top surface. Defining = T T, the differential equation and boundary...

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4.5 KNOWN: PROBLEM Boundary conditions on four sides of a rectangular plate. FIND: Temperature distribution. SCHEMATIC: y q s W T1 T1 0 0 T1 L x ASSUMPTIONS: (1) Two-dimensional, steady-state conduction, (2) Constant properties. ANALYSIS: This problem differs from the one solved in Section 4.2 only in the boundary condition at the top surface. Defining = T T, the differential equation and boundary conditions are 2 2 + 2 =0 x 2 y (0, y) = 0 (L, y) = 0 (x,0) = 0 k y = q s y=W (1a,b,c,d) The solution is identical to that in Section 4.2 through Equation (4.11), nx ny = Cn sin sinh L L n=1 To determine Cn, we now apply the top surface boundary condition, Equation Differentiating (1d). Equation (2) yields (2) Continued.... PROBLEM 4.5 (Cont.) y = y=W n=1 Cn n nx nW sin cosh L L L (3) Substituting this into Equation (1d) results in q s = k n=1 A n sin nx L (4) where An = Cn(n/L)cosh(nW/L). The principles expressed in Equations (4.13) through (4.16) still apply, but now with reference to Equation (4) and Equation (4.14), we should choose nx . Equation (4.16) then becomes f(x) = q /k , g n (x) = sin s L q nx s sin L dx k 0 L 0 L An = sin 2 nx dx L = q 2 (-1) n+1 + 1 s k n Thus Cn = 2 qL (-1) n+1 + 1 s k n 2 2 cosh(nW/L) (5) The solution is given by Equation (2) with Cn defined by Equation (5).
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Seoul National - MAE - Heat Trans
PROBLEM 4.14KNOWN: Tube embedded in the center plane of a concrete slab. FIND: The shape factor and heat transfer rate per unit length using the appropriate tabulated relation, SCHEMATIC:ASSUMPTIONS: (1) Two-dimensional conduction, (2) Steady-state cond
Seoul National - MAE - Heat Trans
PROBLEM 4.15 KNOWN: Dimensions and boundary temperatures of a steam pipe embedded in a concrete casing. FIND: Heat loss per unit length. SCHEMATIC:ASSUMPTIONS: (1) Steady-state conditions, (2) Negligible steam side convection resistance, pipe wall resist
Seoul National - MAE - Heat Trans
PROBLEM 4.16KNOWN: Thin-walled copper tube enclosed by an eccentric cylindrical shell; intervening space filled with insulation. FIND: Heat loss per unit length of tube; compare result with that of a concentric tube-shell arrangement. SCHEMATIC:ASSUMPTI
Seoul National - MAE - Heat Trans
PROBLEM 4.51KNOWN: Square shape subjected to uniform surface temperature conditions. FIND: (a) Temperature at the four specified nodes; estimate the midpoint temperature To, (b) Reducing the mesh size by a factor of 2, determine the corresponding nodal t
Seoul National - MAE - Heat Trans
PROBLEM 4.53KNOWN: Volumetric heat generation in a rectangular rod of uniform surface temperature. FIND: (a) Temperature distribution in the rod, and (b) With boundary conditions unchanged, heat generation rate causing the midpoint temperature to reach 6
Seoul National - MAE - Heat Trans
PROBLEM 5.5 KNOWN: Diameter and initial temperature of steel balls cooling in air. FIND: Time required to cool to a prescribed temperature. SCHEMATIC:ASSUMPTIONS: (1) Negligible radiation effects, (2) Constant properties. ANALYSIS: Applying Eq. 5.10 to a
Seoul National - MAE - Heat Trans
PROBLEM 5.7 KNOWN: The temperature-time history of a pure copper sphere in an air stream. FIND: The heat transfer coefficient between the sphere and the air stream. SCHEMATIC:ASSUMPTIONS: (1) Temperature of sphere is spatially uniform, (2) Negligible rad
Seoul National - MAE - Heat Trans
PROBLEM 5.8KNOWN: Solid steel sphere (AISI 1010), coated with dielectric layer of prescribed thickness and thermal conductivity. Coated sphere, initially at uniform temperature, is suddenly quenched in an oil bath. FIND: Time required for sphere to reach
Seoul National - MAE - Heat Trans
PROBLEM 5.11KNOWN: Configuration, initial temperature and charging conditions of a thermal energy storage unit. FIND: Time required to achieve 75% of maximum possible energy storage. Temperature of storage medium at this time. SCHEMATIC:ASSUMPTIONS: (1)
Seoul National - MAE - Heat Trans
PROBLEM 5.12KNOWN: Diameter, density, specific heat and thermal conductivity of aluminum spheres used in packed bed thermal energy storage system. Convection coefficient and inlet gas temperature. FIND: Time required for sphere to acquire 90% of maximum
Seoul National - MAE - Heat Trans
PROBLEM 5.16 KNOWN: Thickness and properties of furnace wall. Thermal resistance of film on surface of wall exposed to furnace gases. Initial wall temperature. FIND: (a) Time required for surface of wall to reach a prescribed temperature, (b) Correspondin
Seoul National - MAE - Heat Trans
PROBLEM 5.18KNOWN: Diameter, resistance and current flow for a wire. Convection coefficient and temperature of surrounding oil. FIND: Steady-state temperature of the wire. Time for the wire temperature to come within 1C of its steady-state value. SCHEMAT
Seoul National - MAE - Heat Trans
PROBLEM 5.22KNOWN: Metal sphere, initially at a uniform temperature Ti, is suddenly removed from a furnace and suspended in a large room and subjected to a convection process (T, h) and to radiation exchange with surroundings, Tsur. FIND: (a) Time it tak
Seoul National - MAE - Heat Trans
PROBLEM 5.23 KNOWN: Droplet properties, diameter, velocity and initial and final temperatures. FIND: Travel distance and rejected thermal energy. SCHEMATIC:ASSUMPTIONS: (1) Constant properties, (2) Negligible radiation from space. PROPERTIES: Droplet (gi
Seoul National - MAE - Heat Trans
PROBLEM 5.24KNOWN: Initial and final temperatures of a niobium sphere. Diameter and properties of the sphere. Temperature of surroundings and/or gas flow, and convection coefficient associated with the flow. FIND: (a) Time required to cool the sphere exc
Seoul National - MAE - Heat Trans
PROBLEM 5.25KNOWN: Diameter and thermophysical properties of alumina particles. Convection conditions associated with a two-step heating process. FIND: (a) Time-in-flight (ti-f) required for complete melting, (b) Validity of assuming negligible radiation
Seoul National - MAE - Heat Trans
PROBLEM 5.34 KNOWN: Series solution, Eq. 5.39, for transient conduction in a plane wall with convection. FIND: Midplane (x*=0) and surface (x*=1) temperatures * for Fo=0.1 and 1, using Bi=0.1, 1 and 10 with only the first four eigenvalues. Based upon thes
Seoul National - MAE - Heat Trans
PROBLEM 5.35 KNOWN: One-dimensional wall, initially at a uniform temperature, Ti, is suddenly exposed to a convection process (T, h). For wall #1, the time (t1 = 100s) required to reach a specified temperature at x = L is prescribed, T(L1, t1) = 315C. FIN
Seoul National - MAE - Heat Trans
PROBLEM 5.37KNOWN: Thickness, properties and initial temperature of steel slab. Convection conditions. FIND: Heating time required to achieve a minimum temperature of 550C in the slab. SCHEMATIC:ASSUMPTIONS: (1) One-dimensional conduction, (2) Negligibl
Seoul National - MAE - Heat Trans
PROBLEM 5.39KNOWN: Thickness, initial temperature and properties of furnace wall. Convection conditions at inner surface. FIND: Time required for outer surface to reach a prescribed temperature. Corresponding temperature distribution in wall and at inter
Seoul National - MAE - Heat Trans
PROBLEM 5.46KNOWN: Stack of circuit board-pressing plates, initially at a uniform temperature, is subjected by upper/lower platens to a higher temperature. FIND: (a) Elapsed time, te, required for the mid-plane to reach cure temperature when platens are
Seoul National - MAE - Heat Trans
PROBLEM 5.57KNOWN: A ball bearing is suddenly immersed in a molten salt bath; heat treatment to harden occurs at locations with T &gt; 1000 K. FIND: Time required to harden outer layer of 1mm. SCHEMATIC:ASSUMPTIONS: (1) One-dimensional radial conduction, (
Seoul National - MAE - Heat Trans
PROBLEM 5.59KNOWN: Diameter and initial temperature of ball bearings to be quenched in an oil bath. FIND: (a) Time required for surface to cool to 100C and the corresponding center temperature, (b) Oil bath cooling requirements. SCHEMATIC:ASSUMPTIONS: (
Seoul National - MAE - Heat Trans
PROBLEM 5.60KNOWN: Sphere quenching in a constant temperature bath. FIND: (a) Plot T(0,t) and T(ro,t) as function of time, (b) Time required for surface to reach 415 K, t , (c) Heat flux when T(ro, t ) = 415 K, (d) Energy lost by sphere in cooling to T(r
Seoul National - MAE - Heat Trans
PROBLEM 5.66 KNOWN: Two large blocks of different materials like copper and concrete at room temperature, 23C. FIND: Which block will feel cooler to the touch? SCHEMATIC:ASSUMPTIONS: (1) Blocks can be treated as semi-infinite solid, (2) Hand or finger te
Seoul National - MAE - Heat Trans
PROBLEM 5.71KNOWN: Tile-iron, 254 mm to a side, at 150C is suddenly brought into contact with tile over a subflooring material initially at Ti = 25C with prescribed thermophysical properties. Tile adhesive softens in 2 minutes at 50C, but deteriorates ab
Seoul National - MAE - Heat Trans
PROBLEM 5.73 KNOWN: Procedure for measuring convection heat transfer coefficient, which involves melting of a surface coating. FIND: Melting point of coating for prescribed conditions. SCHEMATIC:ASSUMPTIONS: (1) One-dimensional conduction in solid rod (n
Seoul National - MAE - Heat Trans
PROBLEM 5.74KNOWN: Irreversible thermal injury (cell damage) occurs in living tissue maintained at T 48C for a duration t 10s. FIND: (a) Extent of damage for 10 seconds of contact with machinery in the temperature range 50 to 100C, (b) Temperature histor
Seoul National - MAE - Heat Trans
PROBLEM 5.80 KNOWN: Initial temperature of copper and glass plates. Initial temperature and properties of finger. FIND: Whether copper or glass feels cooler to touch. SCHEMATIC:ASSUMPTIONS: (1) The finger and the plate behave as semi-infinite solids, (2)
Seoul National - MAE - Heat Trans
PROBLEM 5.99KNOWN: A 0.12 m thick wall, with thermal diffusivity 1.5 10-6 m2/s, initially at a uniform temperature of 85C, has one face suddenly lowered to 20C while the other face is perfectly insulated. FIND: (a) Using the explicit finite-difference me
Seoul National - MAE - Heat Trans
PROBLEM 6.3 KNOWN: Boundary layer temperature distribution. FIND: Surface heat flux. SCHEMATIC:PROPERTIES: Table A-4, Air (Ts = 300K): k = 0.0263 W/mK. ANALYSIS: Applying Fourier's law at y = 0, the heat flux is q = -k s u y u = - k ( T - Ts ) Pr exp - P
Seoul National - MAE - Heat Trans
PROBLEM 6.6 KNOWN: Expression for the local heat transfer coefficient of a circular, hot gas jet at T directed normal to a circular plate at Ts of radius ro. FIND: Heat transfer rate to the plate by convection. SCHEMATIC:ASSUMPTIONS: (1) Steady-state con
Seoul National - MAE - Heat Trans
PROBLEM 6.20 KNOWN: Experimental measurements of the heat transfer coefficient for a square bar in cross flow. FIND: (a) h for the condition when L = 1m and V = 15m/s, (b) h for the condition when L = 1m and V = 30m/s, (c) Effect of defining a side as the
Seoul National - MAE - Heat Trans
PROBLEM 6.23 KNOWN: Variation of hx with x for flow over a flat plate. FIND: Ratio of average Nusselt number for the entire plate to the local Nusselt number at x = L. SCHEMATIC:ANALYSIS: The expressions for the local and average Nusselt numbers areNu L
Seoul National - MAE - Heat Trans
PROBLEM 6.24 KNOWN: Laminar boundary layer flow of air at 20C and 1 atm having t = 1.13 . FIND: Ratio / t when fluid is ethylene glycol for same conditions. SCHEMATIC:ASSUMPTIONS: (1) Laminar flow. PROPERTIES: Table A-4, Air (293K, 1 atm): Pr = 0.709; Ta
Seoul National - MAE - Heat Trans
PROBLEM 6.29KNOWN: Form of Nusselt number for flow of air or a dielectric liquid over components of a circuit card. FIND: Ratios of time constants associated with intermittent heating and cooling. Fluid that provides faster thermal response. PROPERTIES:
Seoul National - MAE - Heat Trans
PROBLEM 6.35 KNOWN: Air flow conditions and drag force associated with a heater of prescribed surface temperature and area. FIND: Required heater power. SCHEMATIC:ASSUMPTIONS: (1) Steady-state conditions, (2) Reynolds analogy is applicable, (3) Bottom su
Seoul National - MAE - Heat Trans
PROBLEM 6.36 KNOWN: Heat transfer correlation associated with parallel flow over a rough flat plate. Velocity and temperature of air flow over the plate. FIND: Surface shear stress l m from the leading edge. SCHEMATIC:ASSUMPTIONS: (1) Modified Reynolds a
Seoul National - MAE - Heat Trans
PROBLEM 6.44 KNOWN: Species concentration profile, CA(y), in a boundary layer at a particular location for flow over a surface. FIND: Expression for the mass transfer coefficient, hm, in terms of the profile constants, CA, and DAB. Expression for the mola
Seoul National - MAE - Heat Trans
PROBLEM 6.50KNOWN: Convection heat transfer correlation for flow over a contoured surface. FIND: (a) Evaporation rate from a water film on the surface, (b) Steady-state film temperature. SCHEMATIC:ASSUMPTIONS: (1) Steady-state conditions, (b) Constant p
Seoul National - MAE - Heat Trans
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