Assignment on Heat Transfer

ME 303: Convection, Boiling, Condensation and Mass Transfer Assignment #1 Last Date of Submission: 04 November, 2015 before 12.00pm

Submission place: M413

Problem 1: A 6-cm diameter shaft rotates at 3000 rpm in a 20-cm long bearing with a uniform clearance of 0.2 mm. At steady operating conditions, both the bearing and the shaft in the vicinity of the oil gap are at 50°C, and the viscosity and thermal conductivity of lubricating oil are 0.05 N.s/m2 and 0.17 W/m.K. By simplifying and solving the continuity, momentum, and energy equations, determine (a) the maximum temperature of oil, and (b) the rates of heat transfer to the bearing and the shaft.

Problem 2: The forming section of a plastics plant puts out a continuous sheet of plastic that is 1.2 m wide and 2 mm thick at a rate of 15 m/min. The temperature of the plastic sheet is 90°C when it is exposed to the surrounding air, and the sheet is subjected to air flow at 30°C at a velocity of 3 m/s on both sides along its surfaces normal to the direction of motion of the sheet. The width of the air cooling section is such that a fixed point on the plastic sheet passes through that section in 2 s. Determine the rate of heat transfer from the plastic sheet to the air.

Problem 3: An array of power transistors, dissipating 6 W of power each, are to be cooled by mounting them on a 25-cm × 25-cm square aluminum plate and blowing air at 35°C over the plate with a fan at a velocity of 4 m/s. The average temperature of the plate is not to exceed 65°C. Assuming the heat transfer from the back side of the plate to be negligible and disregarding radiation, determine the number of transistors that can be placed on this plate.

Problem 4: Atmospheric air at 375 K flows with a velocity of 4 m/s along a flat plate of 1 m long, maintained at a uniform temperature 325 K. The average heat transfer coefficient is determined to be 8 W/m2°C. Using the Colburn-Reynolds analogy, estimate the drag force acting on the plate over the width of 2 m.

Problem 5: During a plant visit, it was noticed that a 12-m long section of a 10-cm diameter stream pipe is completely exposed to the ambient air. The temperature measurements indicate that the average temperature of the outer surface of the stream pipe is 75°C when the ambient temperature is 5°C. There are also light winds in the area at 10 km/h. The emissivity of the outer surface of the pipe is 0.8, and the average temperature of the surfaces surrounding the pipe, including the sky, is estimated to be 0°C. Determine the amount of heat lost from the steam during a 10-h long work day.

Problem 6: Air flows over a flat plate at a constant velocity of 20 m/s and ambient conditions of 20 kPa and 20°C. The plate is heated to a constant temperature of 75°C, starting at a distance of 7.5 cm from the leading edge. What is the total heat transfer from the leading edge to a point 35 cm from the leading edge? Problem 7: Air flows across a 20 cm square plate with a velocity of 5 m/s. Free-stream conditions are 10°C and 0.2 atm. A heater in the plate surface furnishes a constant heat flux condition at the wall so that the average wall temperature is 100°C. Calculate the surface heat flux and the value of h at an x position of 10 cm. Problem 8: A blackened plate is exposed to the sun so that a constant heat flux of 800 W/m2 is absorbed. The black side of the plate is insulated so that all the energy absorbed is dissipated to an air stream which blows across the plate at conditions of 25°C, 1 bar and 3 m/s. The plate is 25 cm square. Estimate the average temperature of the plate. What is the plate temperature at the trailing edge? Problem 9: Consider a 50-cm diameter and 95-cm long hot water tank. The tank is placed horizontally on the roof of a house. The water inside the tank heated to 80°C by a flat-plate solar collector during the day. The tank is then exposed to windy air at 18°C with an average velocity of 40 km/h during the night. Estimate the temperature of the tank a 45-min period. Assume the tank surface to be at the same temperature as the water inside, and the heat transfer coefficient on the top and bottom surfaces to be the same as that on the side surface. Use the following correlation proposed by Churchill and Bernstein (1977): RedPr > 0.2

Problem 10: Air flows across a 4 cm square cylinder at a velocity of 10 m/s. The surface temperature is maintained at 85°C. Free-stream air conditions are 20°C and 0.6 bar. Calculate the heat loss from the cylinder per meter of length.

Type Local Average Local Average Average

Local

Local

Local

Average

Summary of Correlation for Forced Convection Flow over Flat Plates Properties evaluated at Film temperature Heat Transfer Restrictions Fluid Flow Isothermal (Tw = constant) Isoflux (qw = constant) C f ,x = 0.664Re−x 1/ 2 Nu x = 0.332Re1x/ 2 Pr1/ 3 Nu x = 0.453Re1x/ 2 Pr1/ 3 Laminar: Rex < 5 × 105 ; 0.6 < Pr < 50 Laminar: ReL < 5 × 105 ; 0.6 < Pr < 50 Turbulent: 5 × 105 ≤ Rex ≤ 107 ; 0.6 ≤ Pr ≤ 60 Turbulent: 5 × 105 ≤ ReL ≤ 107 ; 0.6 ≤ Pr ≤ 60 Partly Laminar, Partly Turbulent: 5 × 105 ≤ ReL ≤ 107 ; 0.6 ≤ Pr ≤ 60 Recr = 5 × 105

C f = 1.328Re −L1/ 2

Nu L = 0.664Re1L/ 2 Pr1/ 3

Nu L = 0.680Re1L/ 2 Pr1/ 3

C f ,x = 0.059Re−x 1/ 5

Nu x = 0.0296Re 4x / 5 Pr1/ 3

Nu x = 0.0308Re 4x / 5 Pr1/ 3

C f = 0.074Re−L1/ 5

Nu L = 0.037Re 4L/ 5 Pr1/ 3

Nu L = 0.037Re 4L/ 5 Pr1/ 3

C f = 0.074Re

−1 / 5 L

− 1742Re L

All Prandtl number (Churchill and Ozoe): Pex ≥ 100 ξ = unheated starting length C f ,x = 0.664Re−x 1/ 2 Laminar: Rex < 5 × 105 ; 0.6 < Pr < 50 ξ = unheated starting length C f ,x = 0.059Re−x 1/ 5 Turbulent: 5 × 105 ≤ Rex ≤ 107 ; 0.6 ≤ Pr ≤ 60 ξ = unheated starting length Laminar: ReL < 5 × 105; p = 2 Turbulent: 5 × 105 ≤ ReL ≤ 107; p = 8

Nu L = ( 0.037Re

Nu x =

4/ 5 L

− 871) Pr

1/ 3

0.3387Re1x/ 2 Pr1/ 3   0.0468  1 +     Pr 

2 / 3 1/ 4

  

0.037Re 4L/ 5 Pr1/ 3 Nu L = 1 + 12.35 × 106 Re−L6 / 5 Nu x =

0.4637Re1x/ 2 Pr1/ 3   0.0207  2 / 3  1 +    Pr    

  ξ 3 / 4  Nu x = Nu x( for ξ =0) 1 −      x  

−1 / 3

  ξ 9 / 10  Nu x = Nu x( for ξ =0) 1 −      x  

 L Nu L = Nu L( for ξ =0)   L −ξ

−1 / 9

  ξ   1 −   x  

p +1 p+2

   

p p +1

1/ 4

443 CHAPTER 7

The characteristic length D for use in the calculation of the Reynolds and the Nusselt numbers for different geometries is as indicated on the figure. All fluid properties are evaluated at the film temperature. Note that the values presented in Table 7–1 for non-circular geometrics have been updated based on the recommendations of Sparrow et al. (2004).

TABLE 7–1 Empirical correlations for the average Nusselt number for forced convection over circular and noncircular cylinders in cross flow (from Zukauskas, 1972, Jakob 1949, and Sparrow et al., 2004) Cross-section of the cylinder

Fluid

Circle D

Square

Range of Re

Nusselt number 0.989Re0.330 0.911Re0.385 0.683Re0.466 0.193Re0.618 0.027Re0.805

Pr1/3 Pr1/3 Pr1/3 Pr1/3 Pr1/3

Gas or liquid

0.4–4 4–40 40–4000 4000–40,000 40,000–400,000

Nu Nu Nu Nu Nu

Gas

3900–79,000

Nu 5 0.094Re0.675 Pr1/3

Gas

5600–111,000

Nu 5 0.258Re0.588 Pr1/3

Gas

4500–90,700

Nu 5 0.148Re0.638 Pr1/3

Gas

5200–20,400 20,400–105,000

Nu 5 0.162Re0.638 Pr1/3 Nu 5 0.039Re0.782 Pr1/3

Gas

6300–23,600

Nu 5 0.257Re0.731 Pr1/3

Gas

1400–8200

Nu 5 0.197Re0.612 Pr1/3

5 5 5 5 5

D

Square (tilted 45°)

D

Hexagon D

Hexagon (tilted 45°)

Vertical plate

D

D

Ellipse D

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A P P E N D I X A Tables

Table A-5 Properties of air at atmospheric pressure.† The values of µ, k, cp , and Pr are not strongly pressure-dependent and may be used over a fairly wide range of pressures

T ,K 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

ρ kg/m3

cp kJ/kg · ◦ C

µ × 105 kg/m· s

ν × 106 m2 /s

k W/m · ◦ C

α × 104 m2 /s

Pr

3.6010 2.3675 1.7684 1.4128 1.1774 0.9980 0.8826 0.7833 0.7048 0.6423 0.5879 0.5430 0.5030 0.4709 0.4405 0.4149 0.3925 0.3716 0.3524 0.3204 0.2947 0.2707 0.2515 0.2355 0.2211 0.2082 0.1970 0.1858 0.1762 0.1682 0.1602 0.1538 0.1458 0.1394

1.0266 1.0099 1.0061 1.0053 1.0057 1.0090 1.0140 1.0207 1.0295 1.0392 1.0551 1.0635 1.0752 1.0856 1.0978 1.1095 1.1212 1.1321 1.1417 1.160 1.179 1.197 1.214 1.230 1.248 1.267 1.287 1.309 1.338 1.372 1.419 1.482 1.574 1.688

0.6924 1.0283 1.3289 1.5990 1.8462 2.075 2.286 2.484 2.671 2.848 3.018 3.177 3.332 3.481 3.625 3.765 3.899 4.023 4.152 4.44 4.69 4.93 5.17 5.40 5.63 5.85 6.07 6.29 6.50 6.72 6.93 7.14 7.35 7.57

1.923 4.343 7.490 11.31 15.69 20.76 25.90 31.71 37.90 44.34 51.34 58.51 66.25 73.91 82.29 90.75 99.3 108.2 117.8 138.6 159.1 182.1 205.5 229.1 254.5 280.5 308.1 338.5 369.0 399.6 432.6 464.0 504.0 543.5

0.009246 0.013735 0.01809 0.02227 0.02624 0.03003 0.03365 0.03707 0.04038 0.04360 0.04659 0.04953 0.05230 0.05509 0.05779 0.06028 0.06279 0.06525 0.06752 0.0732 0.0782 0.0837 0.0891 0.0946 0.100 0.105 0.111 0.117 0.124 0.131 0.139 0.149 0.161 0.175

0.02501 0.05745 0.10165 0.15675 0.22160 0.2983 0.3760 0.4222 0.5564 0.6532 0.7512 0.8578 0.9672 1.0774 1.1951 1.3097 1.4271 1.5510 1.6779 1.969 2.251 2.583 2.920 3.262 3.609 3.977 4.379 4.811 5.260 5.715 6.120 6.540 7.020 7.441

0.770 0.753 0.739 0.722 0.708 0.697 0.689 0.683 0.680 0.680 0.680 0.682 0.684 0.686 0.689 0.692 0.696 0.699 0.702 0.704 0.707 0.705 0.705 0.705 0.705 0.705 0.704 0.704 0.702 0.700 0.707 0.710 0.718 0.730

† From Natl. Bur. Stand. (U.S.) Circ. 564, 1955.

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A P P E N D I X A Tables

Table A-9 Properties of water (saturated liquid).†
Note: Grx Pr =

◦F

◦C

32 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 220 240 260 280 300 350 400 450 500 550 600

0 4.44 10 15.56 21.11 26.67 32.22 37.78 43.33 48.89 54.44 60 65.55 71.11 76.67 82.22 87.78 93.33 104.4 115.6 126.7 137.8 148.9 176.7 204.4 232.2 260 287.7 315.6

gβρ 2 cp µk

 x 3 T

cp

ρ

µ

k

kJ/kg · ◦ C

kg/m3

kg/m · s

W/m · ◦ C

4.225 4.208 4.195 4.186 4.179 4.179 4.174 4.174 4.174 4.174 4.179 4.179 4.183 4.186 4.191 4.195 4.199 4.204 4.216 4.229 4.250 4.271 4.296 4.371 4.467 4.585 4.731 5.024 5.703

999.8 999.8 999.2 998.6 997.4 995.8 994.9 993.0 990.6 988.8 985.7 983.3 980.3 977.3 973.7 970.2 966.7 963.2 955.1 946.7 937.2 928.1 918.0 890.4 859.4 825.7 785.2 735.5 678.7

1.79×10−3 1.55 1.31 1.12 9.8×10−4 8.6 7.65 6.82 6.16 5.62 5.13 4.71 4.3 4.01 3.72 3.47 3.27 3.06 2.67 2.44 2.19 1.98 1.86 1.57 1.36 1.20 1.07 9.51×10−5 8.68

0.566 0.575 0.585 0.595 0.604 0.614 0.623 0.630 0.637 0.644 0.649 0.654 0.659 0.665 0.668 0.673 0.675 0.678 0.684 0.685 0.685 0.685 0.684 0.677 0.665 0.646 0.616

Pr 13.25 11.35 9.40 7.88 6.78 5.85 5.12 4.53 4.04 3.64 3.30 3.01 2.73 2.53 2.33 2.16 2.03 1.90 1.66 1.51 1.36 1.24 1.17 1.02 1.00 0.85 0.83

gβρ 2 cp µk 1/m3 · ◦ C

1.91 × 109 6.34 × 109 1.08 × 1010 1.46 × 1010 1.91 × 1010 2.48 × 1010 3.3 × 1010 4.19 × 1010 4.89 × 1010 5.66 × 1010 6.48 × 1010 7.62 × 1010 8.84 × 1010 9.85 × 1010 1.09 × 1011

†Adapted to SI units from A. I. Brown and S. M. Marco, Introduction to Heat Transfer, 3rd ed. New York: McGraw-Hill, 1958.

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