Fundamentals of Thermal-Fluid Sciences
Fundamentals of Thermal-Fluid Sciences
5th Edition
ISBN: 9780078027680
Author: Yunus A. Cengel Dr., Robert H. Turner, John M. Cimbala
Publisher: McGraw-Hill Education
bartleby

Videos

Question
Book Icon
Chapter 18, Problem 61P
To determine

The center temperature of the apples, the surface temperature of the apples and the amount of heat transfer from the apple.

Expert Solution & Answer
Check Mark

Explanation of Solution

Given:

Diameter of the apples (D) is 9cm.

Initial temperature of the apple (Ti) is 20°C.

Temperature of the freezer (T) is 15°C.

Time taken to cool the apples (t) is 1h.

Convection heat transfer coefficient (h) is 8W/m2K.

Calculation:

Write the given properties of the apples.

  ρ=840kg/m3cp=3.81J/kgKk=0.418W/mKα=1.3×107m2/s

Calculate the Biot number (Bi).

  Bi=hr0k=h(D2)k=(8W/m2K)(0.045m)0.418W/mK=0.861

Refer Table 18-2, “Coefficients used in the one-term approximate solution of transient one-dimensional heat conduction in plane walls, cylinders, and spheres”, obtain the constants λ1 and A1 corresponding to the Biot number of 0.861.

  λ1=1.476A1=1.2390

Calculate the Fourier number (τ).

  τ=αtL2=(1.3×107m2/s)(1h)(0.045m)2=(1.3×107m2/s)(1×3600s)(0.045m)2=0.231

The Fourier number is greater than 0.2(0.231>0.2). Therefore, the one-term approximate solution is applicable.

Calculate the temperature at the center of the apples (T0).

  θ0,sph=A1eλ12τT0TTiT=A1eλ12τ

  T0(15°C)20°C(15°C)=(1.2390)e(1.476)2(0.231)=0.749

  T0=11.2°C

Thus, the center temperature of the apples is 11.2°C.

Calculate the temperature at the surface of the apples (T(r0,t)).

  θ(r0,t)sph=A1eλ12τ[sin(λ1r0r0)(λ1r0r0)]T(r0,t)TTiT=A1eλ12τ[sin(λ1r0r0)(λ1r0r0)]

  T(r0,t)(15°C)20°C(15°C)=(1.2390)e(1.476)2(0.231)[sin(1.476)(1.476)]=0.505

  T(r0,t)=2.7°C

Thus, the center temperature of the apples is 2.7°C.

Calculate the mass of the apples (m).

  m=ρV=ρ(43πr03)=(840kg/m3)(43π(0.045m)3)=0.3206kg

Calculate the maximum amount of heat transferred to the apples (Qmax).

  Qmax=mcp(TiT)=(0.3206kg)(3.81J/kgK)(20°C(15°C))=42.8kJ

Calculate the amount of heat transfer from the apple (Q).

  (QQmax)cyl=13(θ0,sph)(sinλ1λ1cosλ1λ13)=13(0.749)(sin(1.476)(1.476)cos(1.476)(1.476)3)=0.402

  Q=(0.402)Qmax=(0.402)(42.8kJ)=17.2kJ

Thus, the amount of heat transfer from the apple is 17.2kJ.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Chapter 18 Solutions

Fundamentals of Thermal-Fluid Sciences

Ch. 18 - Prob. 11PCh. 18 - Prob. 12PCh. 18 - Prob. 13PCh. 18 - Prob. 14PCh. 18 - Consider a 1000-W iron whose base plate is made of...Ch. 18 - Metal plates (k = 180 W/m·K, ρ = 2800 kg/m3, and...Ch. 18 - A 5-mm-thick stainless steel strip (k = 21 W/m·K,...Ch. 18 - A long copper rod of diameter 2.0 cm is initially...Ch. 18 - Prob. 21PCh. 18 - Steel rods (ρ = 7832 kg/m3, cp = 434 J/kg·K, and k...Ch. 18 - Prob. 23PCh. 18 - The temperature of a gas stream is to be measured...Ch. 18 - Prob. 25PCh. 18 - A thermocouple, with a spherical junction diameter...Ch. 18 - Prob. 27PCh. 18 - Prob. 28PCh. 18 - Carbon steel balls (ρ = 7833 kg/m3, k = 54 W/m·K,...Ch. 18 - Prob. 31PCh. 18 - Prob. 32PCh. 18 - Prob. 33PCh. 18 - Prob. 34PCh. 18 - Prob. 35PCh. 18 - Prob. 36PCh. 18 - Prob. 37PCh. 18 - Prob. 38PCh. 18 - Prob. 39PCh. 18 - A body at an initial temperature of Ti is brought...Ch. 18 - In a meat processing plant, 2-cm-thick steaks (k =...Ch. 18 - Prob. 42PCh. 18 - Prob. 43PCh. 18 - Prob. 45PCh. 18 - Prob. 46PCh. 18 - Prob. 47PCh. 18 - Prob. 48PCh. 18 - Prob. 49PCh. 18 - A long iron rod (ρ = 7870 kg/m3, cp = 447 J/kg·K,...Ch. 18 - Prob. 51PCh. 18 - A long 35-cm-diameter cylindrical shaft made of...Ch. 18 - Prob. 54PCh. 18 - Prob. 55PCh. 18 - Prob. 56PCh. 18 - A father and son conducted the following simple...Ch. 18 - Prob. 58PCh. 18 - Prob. 59PCh. 18 - Citrus fruits are very susceptible to cold...Ch. 18 - Prob. 61PCh. 18 - Prob. 63PCh. 18 - Prob. 64PCh. 18 - Prob. 65PCh. 18 - White potatoes (k = 0.50 W/m·K and α = 0.13 × 10−6...Ch. 18 - Prob. 67PCh. 18 - Prob. 68PCh. 18 - Prob. 69PCh. 18 - Consider a hot semi-infinite solid at an initial...Ch. 18 - Prob. 71PCh. 18 - Prob. 72PCh. 18 - Prob. 73PCh. 18 - Prob. 74PCh. 18 - Prob. 75PCh. 18 - Prob. 76PCh. 18 - Prob. 77PCh. 18 - Prob. 78PCh. 18 - Prob. 79PCh. 18 - Prob. 81PCh. 18 - Prob. 82PCh. 18 - Prob. 83PCh. 18 - Prob. 84PCh. 18 - Prob. 85PCh. 18 - Prob. 86PCh. 18 - Prob. 88PCh. 18 - Prob. 89PCh. 18 - A 2-cm-high cylindrical ice block (k = 2.22 W/m·K...Ch. 18 - Prob. 91PCh. 18 - Prob. 93PCh. 18 - Prob. 94RQCh. 18 - Large steel plates 1.0-cm in thickness are...Ch. 18 - Prob. 96RQCh. 18 - Prob. 97RQCh. 18 - Prob. 98RQCh. 18 - Prob. 99RQCh. 18 - Prob. 100RQCh. 18 - Prob. 101RQCh. 18 - Prob. 102RQCh. 18 - The water main in the cities must be placed at...Ch. 18 - Prob. 104RQCh. 18 - Prob. 105RQCh. 18 - Prob. 106RQCh. 18 - Prob. 107RQ
Knowledge Booster
Background pattern image
Mechanical Engineering
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Text book image
Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press
Text book image
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON
Text book image
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Text book image
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Text book image
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Text book image
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY
Heat Transfer – Conduction, Convection and Radiation; Author: NG Science;https://www.youtube.com/watch?v=Me60Ti0E_rY;License: Standard youtube license