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An ordinary egg can be approximated as a 5.5-cm-diameter sphere whose thermal conductivity of roughly W = : 0.6 overall density of p = 1000 k and heat capacity of Cp = 3000; mk m² kgk k I 90°C The egg is initially at a uniform temperature of T; = 10°C and is dropped into boiling water at T Taking the convective heat transfer coefficient to be h = 10. determine how long it will take for the egg to W m²K reach T = 70°C. = In solving this problem, please use the One-term approximation model to compute the time needed by the center of the egg and the surface of the egg to reach the desired temperature of 70 C

An ordinary egg can be approximated as a 5.5-cm-diameter sphere whose thermal conductivity of roughly W = : 0.6 overall density of p = 1000 k and heat capacity of Cp = 3000; mk m² kgk k I 90°C The egg is initially at a uniform temperature of T; = 10°C and is dropped into boiling water at T Taking the convective heat transfer coefficient to be h = 10. determine how long it will take for the egg to W m²K reach T = 70°C. = In solving this problem, please use the One-term approximation model to compute the time needed by the center of the egg and the surface of the egg to reach the desired temperature of 70 C

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
8th Edition
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
Publisher:Kreith, Frank; Manglik, Raj M.
Chapter8: Natural Convection
Section: Chapter Questions
Problem 8.28P
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An ordinary egg can be approximated as a 5.5-cm-diameter sphere whose thermal conductivity of roughly W overall density of p= 1000. mk' kg m² and heat capacity of Cp = 3000; kgk k = 0.6- : The egg is initially at a uniform temperature of T₁ = 10°C and is dropped into boiling water at T = 90°C Taking the convective heat transfer coefficient to be h = 10. - determine how long it will take for the egg to W m²K reach T = 70°C. In solving this problem, please use the One-term approximation model to compute the time needed by the center of the egg and the surface of the egg to reach the desired temperature of 70 C Please discuss the difference between the three results. In solving the problem, please follow the steps below. For the One-term approximation: 1. Compute the Biot number, use the egg radius (not the diameter) as the characteristic length T-Too 2. Compute the non-dimensional excess temperature 0 = Ti-Too 3. Look up in the table the values for A₁ and ₁ for the Biot number found above (the table is shown on the back page) - make sure you use the values for sphere for the closest Bi number 4. For calculation of time needed by the center to get 70°C, first calculate T using 8 = A₁ e 2² and then at calculate the time using the definition of T = Fo = 2 5. For calculation of time needed by the surface to get 70°C, first you need to calculate T using 0 = A₁ e-2³¹ sin (¹¹7) where r = ro for the egg surface. After that, you calculate the time using the definition of Fo= T above TABLE 1 Coefficients used in the one-term approximate solution of transient one- dimensional heat conduction in plane walls, cylinders, and spheres (Bi = huk for a plane wall of thickness 2L, and Bi= hr/k for a cylinder or sphere of radius r.) Cylinder 9.0 10.0 20.0 30.0 40.0 50.0 100.0 Sphere Plane Wall A₁ A₁ Bi A₁ A₁ 0.08 0.1 0.2 0.3 λ₁ A₁ 0.01 0.0998 1.0017 0.1412 1.0025 0.1730 1.0030 0.02 0.1410 1.0033 0.1995 1.0050 0.2445 1.0060 0.04 0.1987 1.0066 0.2814 1.0099 0.3450 1.0120 0.06 0.2425 1.0098 0.3438 1.0148 0.4217 1.0179 0.2791 1.0130 0.3960 1.0197 0.4860 1.0239 0.3111 1.0161 0.4417 1.0246 0.5423 1.0298 0.4328 1.0311 1.0483 0.6170 0.7593 1.0592 0.5218 1.0450 0.7465 1.0712 0.9208 1.0880 0.4 0.5932 1.0580 0.8516 1.0931 1.0528 1.1164 0.5 0.6533 1.0701 0.9408 1.1143 1.1656 1.1441 0.6 0.7051 1.0814 1.0184 1.1345 1.2644 1.1713 0.7 0.7506 1.0918 1.0873 1.1539 1.3525 1.1978 0.8 0.7910 1.1016 1.1490 1.1724 1.4320 1.2236 0.9 0.8274 1.1107 1.2048 1.1902 1.5044 1.2488 1.0 0.8603 1.1191 1.2558 1.2071 1.5708 1.2732 2.0 1.0769 1.1785 1.5995 1.3384 2.0288 1.4793 3.0 1.1925 1.2102 1.7887 1.4191 2.2889 1.6227 4.0 1.2646 1.2287 1.9081 1.4698 2.4556 1.7202 5.0 1.3138 1.2403 1.9898 1.5029 2.5704 1.7870 6.0 1.3496 1.2479 2.0490 1.5253 2.6537 1.8338 7.0 1.3766 1.2532 2.0937 1.5411 2.7165 1.8673 8.0 1.3978 1.2570 2.1286 1.5526 2.7654 1.8920 1.4149 1.2598 2.1566 1.5611 2.8044 1.9106 1.4289 1.2620 2.1795 1.5677 2.8363 1.9249 1.4961 1.2699 2.2880 1.5919 2.9857 1.9781 1.5202 1.2717 2.3261 1.5973 3.0372 1.9898 1.5325 1.2723 2.3455 1.5993 3.0632 1.9942 1.5400 1.2727 2.3572 1.6002 3.0788 1.9962 1.5552 1.2731 2.3809 1.6015 3.1102 1.9990 1.5708 1.2732 2.4048 1.6021 3.1416 2.0000 TABLE 2 The zeroth- and first-order Bessel functions of the first kind n 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 Jo(n) 1.0000 0.9975 0.9900 0.9776 0.9604 0.9385 0.9120 0.8812 0.8463 0.8075 0.7652 0.7196 0.6711 0.6201 0.5669 0.5118 0.4554 0.3980 0.3400 0.2818 0.2239 0.1666 0.1104 0.0555 0.0025 J₁(n) 0.0000 0.0499 0.0995 0.1483 0.1960 0.2423 0.2867 0.3290 0.3688 0.4059 0.4400 0.4709 0.4983 0.5220 0.5419 0.5579 0.5699 0.5778 0.5815 0.5812 0.5767 0.5683 0.5560 0.5399 0.5202 -0.4708 2.6 -0.0968 2.8 -0.1850 -0.4097 3.0 -0.2601 -0.3391 3.2 -0.3202 -0.2613
Transcribed Image Text:An ordinary egg can be approximated as a 5.5-cm-diameter sphere whose thermal conductivity of roughly W overall density of p= 1000. mk' kg m² and heat capacity of Cp = 3000; kgk k = 0.6- : The egg is initially at a uniform temperature of T₁ = 10°C and is dropped into boiling water at T = 90°C Taking the convective heat transfer coefficient to be h = 10. - determine how long it will take for the egg to W m²K reach T = 70°C. In solving this problem, please use the One-term approximation model to compute the time needed by the center of the egg and the surface of the egg to reach the desired temperature of 70 C Please discuss the difference between the three results. In solving the problem, please follow the steps below. For the One-term approximation: 1. Compute the Biot number, use the egg radius (not the diameter) as the characteristic length T-Too 2. Compute the non-dimensional excess temperature 0 = Ti-Too 3. Look up in the table the values for A₁ and ₁ for the Biot number found above (the table is shown on the back page) - make sure you use the values for sphere for the closest Bi number 4. For calculation of time needed by the center to get 70°C, first calculate T using 8 = A₁ e 2² and then at calculate the time using the definition of T = Fo = 2 5. For calculation of time needed by the surface to get 70°C, first you need to calculate T using 0 = A₁ e-2³¹ sin (¹¹7) where r = ro for the egg surface. After that, you calculate the time using the definition of Fo= T above TABLE 1 Coefficients used in the one-term approximate solution of transient one- dimensional heat conduction in plane walls, cylinders, and spheres (Bi = huk for a plane wall of thickness 2L, and Bi= hr/k for a cylinder or sphere of radius r.) Cylinder 9.0 10.0 20.0 30.0 40.0 50.0 100.0 Sphere Plane Wall A₁ A₁ Bi A₁ A₁ 0.08 0.1 0.2 0.3 λ₁ A₁ 0.01 0.0998 1.0017 0.1412 1.0025 0.1730 1.0030 0.02 0.1410 1.0033 0.1995 1.0050 0.2445 1.0060 0.04 0.1987 1.0066 0.2814 1.0099 0.3450 1.0120 0.06 0.2425 1.0098 0.3438 1.0148 0.4217 1.0179 0.2791 1.0130 0.3960 1.0197 0.4860 1.0239 0.3111 1.0161 0.4417 1.0246 0.5423 1.0298 0.4328 1.0311 1.0483 0.6170 0.7593 1.0592 0.5218 1.0450 0.7465 1.0712 0.9208 1.0880 0.4 0.5932 1.0580 0.8516 1.0931 1.0528 1.1164 0.5 0.6533 1.0701 0.9408 1.1143 1.1656 1.1441 0.6 0.7051 1.0814 1.0184 1.1345 1.2644 1.1713 0.7 0.7506 1.0918 1.0873 1.1539 1.3525 1.1978 0.8 0.7910 1.1016 1.1490 1.1724 1.4320 1.2236 0.9 0.8274 1.1107 1.2048 1.1902 1.5044 1.2488 1.0 0.8603 1.1191 1.2558 1.2071 1.5708 1.2732 2.0 1.0769 1.1785 1.5995 1.3384 2.0288 1.4793 3.0 1.1925 1.2102 1.7887 1.4191 2.2889 1.6227 4.0 1.2646 1.2287 1.9081 1.4698 2.4556 1.7202 5.0 1.3138 1.2403 1.9898 1.5029 2.5704 1.7870 6.0 1.3496 1.2479 2.0490 1.5253 2.6537 1.8338 7.0 1.3766 1.2532 2.0937 1.5411 2.7165 1.8673 8.0 1.3978 1.2570 2.1286 1.5526 2.7654 1.8920 1.4149 1.2598 2.1566 1.5611 2.8044 1.9106 1.4289 1.2620 2.1795 1.5677 2.8363 1.9249 1.4961 1.2699 2.2880 1.5919 2.9857 1.9781 1.5202 1.2717 2.3261 1.5973 3.0372 1.9898 1.5325 1.2723 2.3455 1.5993 3.0632 1.9942 1.5400 1.2727 2.3572 1.6002 3.0788 1.9962 1.5552 1.2731 2.3809 1.6015 3.1102 1.9990 1.5708 1.2732 2.4048 1.6021 3.1416 2.0000 TABLE 2 The zeroth- and first-order Bessel functions of the first kind n 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 Jo(n) 1.0000 0.9975 0.9900 0.9776 0.9604 0.9385 0.9120 0.8812 0.8463 0.8075 0.7652 0.7196 0.6711 0.6201 0.5669 0.5118 0.4554 0.3980 0.3400 0.2818 0.2239 0.1666 0.1104 0.0555 0.0025 J₁(n) 0.0000 0.0499 0.0995 0.1483 0.1960 0.2423 0.2867 0.3290 0.3688 0.4059 0.4400 0.4709 0.4983 0.5220 0.5419 0.5579 0.5699 0.5778 0.5815 0.5812 0.5767 0.5683 0.5560 0.5399 0.5202 -0.4708 2.6 -0.0968 2.8 -0.1850 -0.4097 3.0 -0.2601 -0.3391 3.2 -0.3202 -0.2613
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