1. Determine the temperature distribution in a solid plate having uniform heat generation !""" of -1 length L length L,, and variable thermal conductivity $ % $ & '( ) *+,; where * is a constant ( K K ). The left and right side of the plate are maintained at the same constant temperature, T w. 2. A nuclear fuel element of thickness 2L is 2L is covered with a steel cladding of thickness b. Heat generated within the nuclear fuel at a rate q, is removed by a fluid T!, which adjoins one surface and is characterized by a convection coefficient h. The other surface is well insulated and the fuel and the steel have a thermal conductivity of k s and k f respectively. Nuclear Fuel
Steel
Steel
Insulation
L
h, T !
L
b
b x
(a) Obtain an equation for temperature distribution T(x) in T(x) in the nuclear fuel. 2 (b) For k k f = 60W/mK, L 60W/mK, L = = 15 mm, b = 3mm, k s = 15 W/mK, h = 10000W/m K and T ! = 200 -C, what are the largest and the smallest temperature in the fuel element. If the heat is generated 7 3 uniformly at a volumetric rate of q = 2"10 W/m . What are the corresponding locations? 3. A bar of square cross-section connects two metallic structures; both structures are maintained at a temperature 200 °C. The bar, 20 mm " 20 mm, is 100 mm long and is made of mild steel (k = = 0.06 kW/mK). The surroundings are at 20 °C and the heat transfer coefficient between 2 the bar and the surroundings is 0.01 kW/m K. Derive an equation for the temperature distribution along the bar and hence calculate the total heat flow rate from the bar to the surroundings. Write Write down the assumptions, T = = 200 °C
T = = 200 °C T s = 20° C k = = 0.06 kW/m L = L = 100 mm 2 h = 0.01 kW/m K
4. The wall of a liquid-to-gas heat exchanger has a surface area on the liquid side of 1.8 m# (0.6 m" 3.0 m) with a heat transfer coefficient of 255 W/m #K. On the other side of the heat
exchanger wall flows a gas, and the wall has 96 thin rectangular steel fins 0.5 cm thick and 1.25 cm high (k = 3 W/m K) as shown in the accompanying sketch. The fins are 3 m long and the heat transfer coefficient on the gas side is 57 W/m # K. Assuming that the thermal resistance of the wall is negligible, determine the rate of heat transfer if the overall temperature difference is 38°C.
5. Derive the steady state temperature distribution in the rectangular p late as shown in the figure 2 and evaluate the temperature at point A, B, C respectively if T = 50 -C, q =1000 W/m , l =1 m, b = 2 m. Point B is located at the center of the rectangle, black strips in the figure shows the insulation. Thermal conductivity of the material is k = 0.06 kW/m.K. b T l/4 b/4 A b/4
l B
C
q 6. A solid conducting body of initial temperature T 1,0 is immersed suddenly in an amount of incompressible liquid of initial temperature T 2,0. The respective heat capacities of the immersed body and the liquid are (mc)1 and (mc)2, where m and c = mass and specific heat capacity of the two entities respectively. The external area of the immersed body is A, and the heat transfer coefficient (h) between two bodies is constant. Treating both the body and the liquid as two lumped capacitance systems, show that their respective temperature vary accordingly to the relation, 2
+. % +./0 1
23/4 526/4
+= % +=/0 )
( 1 : 5;<
'89, .7'89,6 3 23/4 726/4 '89, .7'89,6 3 'AB,3 7'AB,6
where > % ?@
( 1 : 5;< (check the correctness of this expression)
'AB,3 'AB,6
7. A two-dimensional rectangular plate is subjected to the bou ndary conditions shown. Derive an expression for the steady-state temperature distribution T(x, y).
T = T ! H
T = T !
2
T = Ay
L
T = T !
8. The figure shows the triangular cross section through a long bar. A finite temperature difference (" b) is maintained between the two sides that are mutually perpendicular. The hypotenuse is perfectly insulated. Determine analytically the temperature distribution for steady conduction in the rectangular area, " (x,y). (Hint: Exploit the geometrical relationship that might exist between the given triangle a nd a square cross section of length L.) y L
Insulated " = " b
x " = 0
L
9. The figure shows the rectangular cross section of a long bar. Two adjoining sides are at the same temperature T c, while the remaining two sides are at different temperatures, T a and T b. 3
Determine the steady state temperature distribution inside the rectangular cross section. If H = L, what will be the temperature distribution? L T c
H
T c
T b
T a 3
10. A 3 cm diameter aluminum sphere (k = 204 W/m.K, # = 2700 kg/m and c = 0.896 kJ/kg K) is initially at 175 °C. It is suddenly immersed in a well stirred fluid at 25 °C. The temperature of the sphere is lowered to 100 °C in 42 s. Calculate the heat transfer coefficient. 11. A small aluminum sphere of diameter D, initially at a uniform temperature To, is immersed in a liquid whose temperature, T !, varies sinusoidally according to
+C 1 +& % @DE> FG where T m = time-averaged temperature of the liquid, A = amplitude of the temperature fluctuation, $ = frequency of the fluctuations. If the heat transfer coefficient between the fluid and the sphere, h0 , is constant and the system can be treated as a lumped capacity, derive an expression for the sphere temperature as a function of time. 12. A square, 2–D rod is exposed to identical convection conditions on the left and right faces. The bottom surface is insulated, and the top surface receives a non-uniform heat flux given by, ""
! H
% !&I JKL
1M
=
N H1 O
=
where !&I and a are constants. Formulate the problem for the temperature distribution in appropriate dimensionless form, and derive a solution using the SOV method. Note: the boundary conditions can be simplified by exploiting the symmetry of the problem. 13. A long solid circular cylinder of radius R is initially at a temperature of T!. At t = 0, one side of the cylindrical surface is exposed to a source of thermal radiation which results in a heat flux into the cylinder. The distribution of flux is given by,
U U 1 V T V O O U XU V T V O O
!PI % ! &I QRST !PI % W 4
(a) Formulate the problem in dimensionless variables and parameters. (b) Devise a partial solutions approach to the problem, involving a transient solution and a steady–state solution. (c) Derive the transient and steady parts to the solution. 14. A long square rod, of width 30 cm, is to be heated in a high–temperature convection furnace. The rod is at an initial temperature of 30 C and has a thermal conductivity and thermal 6 2 diffusivity of 60 W/m.K and 18"10% m /s, respectively. The rod will remain in the furnace until the center temperature reaches 300 C. To minimize thermal stresses in the rod, the surface temperature of the rod cannot exceed 600 C during the heating. The objective of this problem is to develop a pair of design curves for the furnace. One plot would give hmax vs. oven temperature T!, where hmax is the maximum allowable heat transfer coefficient, which maintains the surface temperature constraint. For example, if T! & 600 C then hmax would ' ! – since the surface temperature could not exceed 600 C for this condition. For any T! greater than 600 C, the value of hmax will be finite, and will decrease as T! increases. The second plot would show the required heating time t as a function of T!. Considering that h 2 values for forced convection in air are around 100–1000 W/m .K, comment on the likely operating conditions for the furnace. °
°
°
°
°
°
15. For the problem shown in the figure below, find the steady temperature distribution. Boundary condition is mixed at x=0 and x=1 insulated. For y=0 convective and y=1 up to x=x1 convective and x>x1 insulated. Width of problem W .
16. In a large concrete slab, the temperature distribution across the thickness of 60 cm, heated from one side as measured by thermocouples approximately to the following relation 2 3 4 o T=50+50x-12x +15x -15x , where T is in C and x is in meters. Considering an area of 5 2 m . Compute (a) Heat entering and leaving the slab in unit time (b) Heat energy stored/unit time (c) The point whose the rate of heating or cooling is maximum (d) the rate of temperature change at both sides of the slab. Take k for slab 1.2 W/mK and Thermal -3 2 diffusivity as 1.77 ! 10 m /h. 1) Consider one-dimensional transient conduction in a plate of width 2L which is initially at a specified temperature distribution given by T i=T(x, 0) =f(x). The plate is suddenly allowed to exchange heat by convection with an ambient fluid at T ! as shown in figure
5
below. The convection coefficient is h and the thermal diffusivity is $. Assume that f(x) is symmetrical about the center plane; determine the transient temperature of the plate.
17. A very short pulse of high intensity current is passed through a thin wire buried in a thick fiberglass insulation layer. As a result, the wire generates (almost instantaneously) heat source Q% per meter of its length. Find the followings: a. Derive the temperature distribution in the insulation. Show all the steps clearly. b. Determine the time t when the maximum temperature occurring at any distance r from the wire. c. What is the maximum temperature at that location?
6
