CSU MECH 344 - HW2 - Solutions (9 pages)

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HW2 - Solutions



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HW2 - Solutions

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Pages:
9
School:
Colorado State University- Fort Collins
Course:
Mech 344 - Heat and Mass Transfer

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PROBLEM 2 14 KNOWN Two dimensional body with specified thermal conductivity and two isothermal surfaces of prescribed temperatures one surface A has a prescribed temperature gradient FIND Temperature gradients T x and T y at the surface B SCHEMATIC ASSUMPTIONS 1 Two dimensional conduction 2 Steady state conditions 3 No heat generation 4 Constant properties ANALYSIS At the surface A the temperature gradient in the x direction must be zero That is T x A 0 This follows from the requirement that the heat flux vector must be normal to an isothermal surface The heat rate at the surface A is given by Fourier s law written as T W K 10 2m 30 600W m y A m K m q y A k w A On the surface B it follows that T y B 0 in order to satisfy the requirement that the heat flux vector be normal to the isothermal surface B Using the conservation of energy requirement Eq 1 12c on the body find q y A q x B 0 or q x B q y A Note that q x B k w B T x B and hence T x B q y A k wB 600 W m 10 W m K 1m 60 K m COMMENTS Note that in using the conservation requirement q in q y A and q out q x B PROBLEM 2 17 KNOWN Electrical heater sandwiched between two identical cylindrical 30 mm dia 60 mm length samples whose opposite ends contact plates maintained at To FIND a Thermal conductivity of SS316 samples for the prescribed conditions A and their average temperature b Thermal conductivity of Armco iron sample for the prescribed conditions B c Comment on advantages of experimental arrangement lateral heat losses and conditions for which T1 T2 SCHEMATIC ASSUMPTIONS 1 One dimensional heat transfer in samples 2 Steady state conditions 3 Negligible contact resistance between materials PROPERTIES Table A 2 Stainless steel 316 T 400 K k ss 15 2 W m K Armco iron T 380 K k iron 67 2 W m K ANALYSIS a For Case A recognize that half the heater power will pass through each of the samples which are presumed identical Apply Fourier s law to a sample q kA c k T x 0 5 100V 0 353A 0 015 m q x 15 0 W m K 2 A c T 0 030 m 4 25 0 C The total temperature drop across the length of the sample is T1 L x 25 C 60 mm 15 mm 100 C Hence the heater temperature is Th 177 C Thus the average temperature of the sample is T To Th 2 127 C 400 K We compare the calculated value of k with the tabulated value see above at 400 K and note the good agreement b For Case B we assume that the thermal conductivity of the SS316 sample is the same as that found



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