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
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Chapter 18, Problem 64P

a)

To determine

The heat transfer coefficient at the surface of the rib.

a)

Expert Solution
Check Mark

Explanation of Solution

Given:

Mass of the rib (m) is 3.2kg.

Initial temperature (Ti) is 4.5°C.

Temperature maintained at the oven (T) is 163°C.

Temperature at the center of the meat (T0) is 60°C.

Time taken to roast the rib (t) is 2h 45min.

Calculation:

Write the given properties of the rib.

  ρ=1200kg/m3cp=4.1J/kgKk=0.45W/mKα=0.914×107m2/s

Calculate the radius of the roast (r0).

  m=ρVV=mρ43πr03=mρ

  r0=(3m4πρ)13=(3(3.2kg)4π(1200kg/m3))13=0.08603m

Calculate the Fourier number (τ).

  τ=αtr02=(0.91×107m2/s)(2h 45min)(0.08603m)2=(0.91×107m2/s)(2×3600s+45×60s)(0.08603m)2=0.1217

The Fourier number is nearly close to 0.2(0.12170.2). Therefore, the one-term approximate solution can be applicable.

Calculate the dimensionless temperature of the roast (θ0,sph).

  θ0,sph=A1eλ12τT0TTiT=A1eλ12τ60°C163°C4.5°C163°C=A1eλ12(0.1217)        (I)

Solve Equation (I) by trial and error method using Table 18-2, Coefficients used in the one-term approximate solution of transient one-dimensional heat conduction in plane walls, cylinders, and spheres”.

Equation (I) is satisfied when the Biot number, Bi=30. It corresponds to the constants values given as follows:

  λ1=3.0372A1=1.9898

Calculate the heat transfer coefficient at the surface of the rib (h).

  Bi=hr0k

  h=k(Bi)r0=(0.45W/mK)(30)(0.08603m)=156.9W/m2K

Thus, the heat transfer coefficient at the surface of the rib is 156.9W/m2K.

b)

To determine

The temperature at the surface of the rib.

b)

Expert Solution
Check Mark

Explanation of Solution

Calculation:

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

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

  T(r0,t)163°C4.5°C163°C=(1.9898)e(3.0372)2(0.1217)[sin(3.0372)(3.0372)]=0.0222

  T(r0,t)=159.5°C

Thus, the temperature at the surface of the rib is 159.5°C.

c)

To determine

The amount of heat transferred to the rib.

c)

Expert Solution
Check Mark

Explanation of Solution

Calculation:

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

  Qmax=mcp(TiT)=(3.2kg)(4.1J/kgK)(163°C4.5°C)=2080kJ

Calculate the amount of heat transferred to the rib (Q).

  (QQmax)cyl=13(θ0,sph)(sinλ1λ1cosλ1λ13)=13(0.65)(sin(3.0372)(3.0372)cos(3.0372)(3.0372)3)=0.783

  Q=(0.783)Qmax=(0.783)(2080kJ)=1629kJ

Thus, the amount of heat transferred to the rib is 1629kJ.

d)

To determine

The time taken to cook the medium-done rib.

d)

Expert Solution
Check Mark

Explanation of Solution

Calculation:

It is given that the innermost temperature of the rib is 71°C.

  T0=71°C

Calculate the Fourier number (τ).

  θ0,sph=A1eλ12τT0TTiT=A1eλ12τ71°C163°C4.5°C163°C=(1.9898)e(3.0372)2ττ=0.1336

Calculate the time taken to cook the medium-done rib (t).

  t=τr02α=(0.1336)(0.08603m)20.91×107m2/s=10,866s

  =181min3hr

Thus, the time taken to cook the medium-done rib is 3hr.

The calculated cooking time (3hr) is nearer to the given time of 3hr20min. The difference between the two values is because of the Fourier number being less than 0.2 and therefore, the error in the one-term approximation method.

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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
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