Practice Problems Set – 1 MEC301: Heat Transfer

Q.1 The slab shown in the figure is embedded on five sides in insulation materials. The sixth side is exposed to an ambient temperature through a heat transfer coefficient. Heat is generated in the slab at the rate of 1.0 kW/m3. The thermal conductivity of the slab is 0.2 W/m-K. (a) Solve for the temperature distribution in the slab, noting any assumptions you must make. Be careful to clearly identify the boundary conditions. (b) Evaluate T at the front and back faces of the slab. (c) Show that your solution gives the expected heat fluxes at the back and front faces.

Q.2

Compute overall heat transfer coefficient U for the slab shown in the figure.

Given: Ls = 2 mm = 0.002 m Lc = 3 mm = 0.003 m ks = 17 W/m-K kc = 372 W/m-K Q.3 A 4 mm diameter spherical ball at 50oC is covered by a 1 mm thick plastic insulation (k = 0.13 W/m-K). The ball is exposed to a medium at 15oC, with a combined convection and radiation heat transfer coefficient of 20 W/m2-K. Determine if the plastic insulation on the ball will help or hurt heat transfer from the ball. Q.4 Prove that if k varies linearly with T in a slab, and if heat transfer is one-dimensional and steady, then q may be evaluated precisely using k evaluated at the mean temperature in the slab.

Q.5 Layers of equal thickness of spruce and pitch pine are laminated to make an insulating material. How should the laminations be oriented in a temperature gradient to achieve the best effect? Given: kspruce = 0.11 W/m-K ; kpine = 0.14 W/m-K. Q.6 Consider the composite wall shown in figure. The concrete and brick sections are of equal thickness. Determine T1, T2, q, and the percentage of q that flows through the brick. To do this, approximate the heat flow as one-dimensional. Draw the thermal circuit for the wall and identify all four resistances before you begin.

Q.7 A furnace wall slab is constructed with fire clay of thickness 90 mm (L1) inside and red brick of thickness 450 mm (L2) outside. The temperatures inside and outside the furnace wall are 1100°C (T1) and 30°C (T3), respectively. The thermal conductivity of the red brick (K2) is 0.8W/m-K and that of the fire clay (K1) is 0.3(1 + 0.001T) W/m-K where T is the temperature of the clay in degrees Celsius. Assuming unit surface area, find the conductive heat loss through the furnace wall and the temperature at the interface of the fire clay and red brick. Q.8 A 20 mm diameter copper pipe is used to carry heated water; the external surface of the pipe is subjected to a convective heat transfer coefficient of h = 6 W/m2-K, find the heat loss by convection per meter length of the pipe when the external surface temperature is 80oC and the surroundings are at 20oC. Assuming black body radiation then what is the heat loss by radiation?

Q.9 A plate 0.3 m long and 0.1 wide, with a thickness of 12 mm is made from stainless steel (k = 16 W/m-K), the top surface is exposed to an airstream of temperature 20oC. In an experiment, the plate is heated by an electrical heater (also 0.3 m by 0.1 m) positioned on the underside of the plate and the temperature of the plate adjacent to the heater is maintained at 100oC. A voltmeter and ammeter are connected to the heater and these read 200 V and 0.25 A, respectively. Assuming that the plate is perfectly insulated on all sides except the top surface, what is the convective heat transfer coefficient?

Q.10 Heat flows steadily through a stainless steel wall of thickness Lss = 0.06 m, with a variable thermal conductivity of kss = 1.67 + 0.0143T (C). It is partially insulated on the right side with glass wool of thickness Lgw = 0.1 m, with a thermal conductivity of kgw = 0.04. The temperature on the left-hand side of the stainless steel is 400 C and on the right-hand side of the glass wool is 100 C. Evaluate q and Ti (interface). Q.11 The resistance of a thick cylindrical layer of insulation must be increased. Will Q be lowered more by a small increase of the outside diameter or by the same decrease in the inside diameter? Q.12 Show that the differential equation governing conduction heat transfer in a solid sphere with heat generation is given by

d 2T 2 dT q ”’ 0 , where T is the temperature at any radius r, q’’’ is the heat dr 2 r dr k

generated per unit volume and k is the thermal conductivity of the solid sphere. Show the general nature of the temperature distribution in this case. Q.13 A steel pipe having internal diameter of 2 cm, outer diameter of 2.4 cm and thermal conductivity of steel of 54 W/m-K carries hot water at 95oC. Heat transfer coefficient between the inner surface of steel pipe and the hot water is 600 W/m2-K. An asbestos insulation with thermal conductivity of 0.2 W/m-K and thickness 2 cm is put on the steel pipe. Heat is lost from the outer surface of the asbestos insulated pipe to the surrounding air at 30oC, heat transfer coefficient for the outer surface of the insulation being 8 W/m2-K. Determine: (i) The rate of heat transfer per meter length of the pipe. (ii) Determine the temperature at the inner, outer surfaces of the steel pipe and the outer surface of the insulation. (iii) What do you understand by the term “critical radius of insulation”? What is the value of critical radius in the above question? What is the rate of heat loss, if thickness of insulation were to correspond to critical radius? Comment on the results.

ANSWERS

Q.1 (a)

(b) Tf =27oC; Tb = 22oC Q.2 Q.3 Q.5 Q.6 Q.7 Q.8 Q.9 Q.10 Q.11 Q.12 4109 W/m2-C will help (increase) the heat transfer. Use series combination T1 = 267oC; T2 = 220oC; q = 454.3 W; 47.6 % q = 1498.93 W; T2 = 873.15oC Qconv = 22.6 W/m; Qrad = 29.1 W/m2 h = 12.7 W/m2-K Ti = 398.6 oC; q = 172.3 W/m2 Q will be lowered more by a small increase of the outside diameter. 2 q ”’ 2 T Tw ( R r ) ; Temperature distribution is parabolic in nature. 6k

Q.13 (i) 45.46 W/m

(ii) T1 = 93.774oC; T2 = 93.52oC; T3 = 58.01oC

(iii) 0.025 m