# Heat 4e SM Chap06

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PROPRIETARY MATERIAL. 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

6-1

Solutions Manual for

Heat and Mass Transfer: Fundamentals & Applications Fourth Edition

Yunus A. Cengel & Afshin J. Ghajar McGraw-Hill, 2011

Chapter 6 FUNDAMENTALS OF CONVECTION

PROPRIETARY AND CONFIDENTIAL This Manual is the proprietary property of The McGraw-Hill Companies, Inc. (McGraw-Hill) and protected by copyright and other state and federal laws. By opening and using this Manual the user agrees to the following restrictions, and if the recipient does not agree to these restrictions, the Manual should be promptly returned unopened to McGraw-Hill: This Manual is being provided only to authorized professors and instructors for use in preparing for the classes using the affiliated textbook. No other use or distribution of this Manual is permitted. This Manual may not be sold and may not be distributed to or used by any student or other third party. No part of this Manual may be reproduced, displayed or distributed in any form or by any means, electronic or otherwise, without the prior written permission of McGraw-Hill.

PROPRIETARY MATERIAL. 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

6-2

Mechanisms and Types of Convection

6-1C A fluid flow during which the density of the fluid remains nearly constant is called incompressible flow. A fluid whose density is practically independent of pressure (such as a liquid) is called an incompressible fluid. The flow of compressible fluid (such as air) is not necessarily compressible since the density of a compressible fluid may still remain constant during flow.

6-2C In forced convection, the fluid is forced to flow over a surface or in a tube by external means such as a pump or a fan. In natural convection, any fluid motion is caused by natural means such as the buoyancy effect that manifests itself as the rise of the warmer fluid and the fall of the cooler fluid. The convection caused by winds is natural convection for the earth, but it is forced convection for bodies subjected to the winds since for the body it makes no difference whether the air motion is caused by a fan or by the winds.

6-3C If the fluid is forced to flow over a surface, it is called external forced convection. If it is forced to flow in a tube, it is called internal forced convection. A heat transfer system can involve both internal and external convection simultaneously. Example: A pipe transporting a fluid in a windy area.

6-4C The convection heat transfer coefficient will usually be higher in forced convection since heat transfer coefficient depends on the fluid velocity, and forced convection involves higher fluid velocities.

6-5C The potato will normally cool faster by blowing warm air to it despite the smaller temperature difference in this case since the fluid motion caused by blowing enhances the heat transfer coefficient considerably.

6-6C Nusselt number is the dimensionless convection heat transfer coefficient, and it represents the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer. It is defined as

khL

Nu c= where Lc is the characteristic length of the surface and k is the thermal conductivity of the fluid.

6-7C Heat transfer through a fluid is conduction in the absence of bulk fluid motion, and convection in the presence of it. The rate of heat transfer is higher in convection because of fluid motion. The value of the convection heat transfer coefficient depends on the fluid motion as well as the fluid properties. Thermal conductivity is a fluid property, and its value does not depend on the flow.

PROPRIETARY MATERIAL. 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

6-3

6-8 Heat transfer coefficients at different air velocities are given during air cooling of potatoes. The initial rate of heat transfer from a potato and the temperature gradient at the potato surface are to be determined.

Assumptions 1 Steady operating conditions exist. 2 Potato is spherical in shape. 3 Convection heat transfer coefficient is constant over the entire surface.

Properties The thermal conductivity of the potato is given to be k = 0.49 W/m.C. Air V = 1 m/s T = 5C

Potato Ti = 20C

Analysis The initial rate of heat transfer from a potato is 222 m 02011.0m) 08.0( === DAs

W5.8=== C5))(20m C)(0.02011. W/m1.19()( 22TThAQ ss& where the heat transfer coefficient is obtained from the table at 1 m/s velocity. The initial value of the temperature gradient at the potato surface is

C/m 585 ===

=

==

=

=

C W/m.49.0C5)C)(20. W/m1.19()(

)(

2

condconv

kTTh

rT

TThrTkqq

s

Rr

sRr

&&

6-9 The rate of heat loss from an average man walking in still air is to be determined at different walking velocities.

Assumptions 1 Steady operating conditions exist. 2 Convection heat transfer coefficient is constant over the entire surface.

Analysis The convection heat transfer coefficients and the rate of heat losses at different walking velocities are

(a) C. W/m956.5m/s) 5.0(6.86.8 20.5353.0 === Vh

Ts = 30C

Air V T = 7C

W246.6=== C)7)(30m C)(1.8. W/m956.5()( 22TThAQ ss& (b) C. W/m60.8m/s) 0.1(6.86.8 20.5353.0 === Vh

W356.0=== C)7)(30m C)(1.8. W/m60.8()( 22TThAQ ss& (c) C. W/m66.10m/s) 5.1(6.86.8 20.5353.0 === Vh W441.3=== C)7)(30m C)(1.8. W/m66.10()( 22TThAQ ss&

(d) C. W/m42.12m/s) 0.2(6.86.8 20.5353.0 === Vh W514.2=== C)7)(30m C)(1.8. W/m42.12()( 22TThAQ ss&

6-4

6-10 The upper surface of a solid plate is being cooled by water. The water convection heat transfer coefficient and the water temperature gradient at the upper plate surface are to be determined.

Assumptions 1 Steady operating conditions exist. 2 Properties are constant. 3 Heat conduction in solid is one-dimensional. 4 No-slip condition at the plate surface.

Properties The thermal conductivity of the solid plate is given as k = 237 W/mK. The thermal conductivity of water at the film temperature of Tf = (Ts,1 + T)/2 = (60C + 20C)/2 = 40C is kfluid = 0.631 W/mK (from Table A-9).

Analysis Applying energy balance on the upper surface of the solid plate (x = 0), we have

convcond qq && = )( 1,1,2, =

TThL

TTk s

ss

The convection heat transfer coefficient for the water is

K W/m711 2 =

=

=

206060120

m 50.0K W/m237

1,

1,2,

TTTT

Lkh

s

ss

The temperature gradient at the upper plate surface (x = 0) for the water is

=

=TT

yTkh

s

y 0fluid )/(

K/m 45,071===

=K )2060(

K W/m631.0K W/m711)(

2

1,fluid0

TTk

hyT

sy

Discussion The film temperature is used to evaluate the thermal conductivity of water (kfluid). This is to account for the effect of temperature on the thermal conductivity.

6-5

6-11 Airflow over a plate surface has a given temperature profile. The heat flux on the plate surface and the convection heat transfer coefficient of the airflow are to be determined.

Assumptions 1 Steady operating conditions exist. 2 Properties are constant. 3 No-slip condition at the plate surface. 4 Heat transfer by radiation is negligible.

Properties The thermal conductivity and thermal diffusivity of air at the film temperature of Tf = (Ts + T)/2 = (220C + 20C)/2 = 120C are kfluid = 0.03235 W/mK and fluid = 3.565 105 m2/s (from Table A-15). Analysis For no-slip condition, heat flux from the solid surface to the fluid layer adjacent to the surface is

24 W/m101.452====

=)C/m 10488.4)(K W/m03235.0( 5

0fluidcond

yyTkqq &&

where the temperature gradient at the plate surface is

C/m 10488.4

/sm 10565.3m/s 08.0)C220C20()(

exp)(

5

25fluid

0fluidfluid0

=

=

=

=

=

=

VTT

yVVTTyT

s

ys

y

The convection heat transfer coefficient of the airflow is

K W/m72.6 2 ==

=

=)C20C220(

)C/m 10488.4)(K W/m03235.0()/( 50fluidTT

yTkh

s

y

Discussion The positive heat flux means that the plate is being cooled by the airflow that passes over the surface of the plate.

6-12 The rate of heat loss from an average man in windy air is to be determined at different wind velocities.

Assumptions 1 Steady operating conditions exist. 2 Convection heat transfer coefficient is constant over the entire surface.

Analysis The convection heat transfer coefficients and the rate of heat losses at different wind velocities are

Ts = 29C 0C