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 17, Problem 127RQ

(a)

To determine

The heat transfer rate for the infinitely long fin.

(a)

Expert Solution
Check Mark

Explanation of Solution

Given:

Thermal conductivity of the fins (k) is 240 W/mK.

Heat transfer coefficient (h) is 250 W/m2K.

Length of the fin (L) is 50 mm.

Diameter of the fin (D) is 10 mm.

Calculation:

Determine the perimeter of the circular fin.

  p=πD=π(0.01 m)=0.03142m

Determine the cross section area of the circular fin.

  Ac=πD24=π(0.01 m)24=7.854×105m2

Determine the value of m.

  m=hpkAc=(250 W/m2°C)(0.03142 m)(240 W/m°C)(7.854×105m2)=20.41 m1

Determine the following factor.

  hpkAc=(250 W/m2°C)(0.03142m)(240 W/m°C)(7.854×105m2)=0.3848 W/°C

Determine the heat transfer rate for an infinitely long fin.

  Q˙long fin =hpkAc(TbT)=(0.3848 W/°C)(350°C25°C)=125 W

Thus, the heat transfer rate for the infinitely long fin is 125 W_.

Write the expression of the temperature variation along the fin.

  T(x)TTbT=emx

(b)

To determine

The heat transfer rate for the adiabatic fin tip.

(b)

Expert Solution
Check Mark

Explanation of Solution

Determine the heat transfer rate for an adiabatic fin tip.

  Q˙adiabatic tip=hpkAc(TbT)tanhmL=(0.3848 W/°C)(350°C25°C)tanh[(20.41 m1)(0.05 m)]=96.3 W

Thus, the heat transfer rate for the adiabatic fin tip is 96.3 W_.

Write the expression of the temperature variation along the fin.

  T(x)TTbT=coshm(Lx)coshmL

(c)

To determine

The heat transfer rate for the fin with tip temperature of 250°C.

(c)

Expert Solution
Check Mark

Explanation of Solution

Determine the heat transfer rate for fin with tip temperature of 250°C.

  Q˙specificad temp =hpkAc(TbT)coshmL(TLT)(TbT)sinhmL=(0.3848 W/°C)(350°C25°C)(0.725)=90.7 W

Thus, the heat transfer rate for the fin with tip temperature of 250°C is 90.7 W_.

Write the expression of the temperature variation along the fin.

  T(x)TTbT=(TLT)(TbT)sinhmx+sinhm(Lx)sinhmL

(d)

To determine

The heat transfer rate for fin with convection from the fin tip.

(d)

Expert Solution
Check Mark

Explanation of Solution

Determine the heat transfer rate for fin with convection from the tip.

  Q˙conv tip =hpkAc(TbT)sinhmL+(hmk)coshmLcoshmL+(hmk)sinhmL=(0.3848 W/°C)(350°C25°C)(0.7901)=98.8 W

Thus, the heat transfer rate for fin with convection from the fin tip is 98.8 W_.

Write the expression of the temperature variation along the fin.

  T(x)TTbT=coshm(Lx)+(hmk)sinhm(Lx)coshmL+(hmk)sinhmL

Tabulate the values of the temperature variation of the single fin.

L, m

T(x)°C
Part (a)Part (b)Part (c)Part (d)
0350350350350
0.005318326328325
0.01290305308304
0.015264288292285
0.02241272279270
0.025220260268256
0.03201250259246
0.035184242253237
0.04169237250231
0.045155233249227
0.05142232250224

Plot the temperature variation of the single fin as in Figure (1).

Fundamentals of Thermal-Fluid Sciences, Chapter 17, Problem 127RQ

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Chapter 17 Solutions

Fundamentals of Thermal-Fluid Sciences

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