mech14.weeblymech14.weebly.com/uploads/6/1/0/6/61069591/me_chmt... · stress vector at a point on a plane normal to n.Using the Gauss divergence theorem and your answer to part (a),

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INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR

Date: 22-02-2012 AN Time: 2 Hours Full Marks: 60

No. of Students: 50 Mid Autumn Semester Examination 2012

Sub. No.: ME60014/ME61004 Sub. Name: Convective Heat and Mass Transfer

Instructions: Attempt all questions. Symbols have their usual meanings. Please explain your work

carefully. Clearly indicate the coordinate system used in your analysis. Make suitable assumptions

wherever necessary. Please state your assumptions clearly.

The following information may be useful:

The divergence theorem implies that j

j j

jB R

aa n d A dV

x

∂=

∂∫ ∫ , where 1 2 3( , , , )x x x ta is a continuous vector field in a region R bounded by a surface B.

The divergence of a vector field, r zv v vφ φ= + +r zv e e e , in cylindrical coordinates is given by

1 1( ) zr

v vrv

r r r z

φ

φ∂∂ ∂

∇ ⋅ = + +∂ ∂ ∂

v .

The gradient, Laplacian and material derivative of a scalar function, ( , , , )T r z tφ , in cylindrical coordinates, are given by

1T T TT

r r zφφ

∂ ∂ ∂∇ = + +

∂ ∂ ∂r ze e e ,

2 22

2 2 2

1 1T T TT r

r r r r zφ∂ ∂ ∂ ∂⎛ ⎞∇ = + +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

,

r z

vDT T T T Tv v

Dt t r r z

φ

φ∂ ∂ ∂ ∂

≡ + + +∂ ∂ ∂ ∂

.

The viscous dissipation function in cylindrical coordinates is given by 2 2 22 2 2

1 1 12 2 2r r z z r z r

v v v vv v v v v v v

r r r z r z r z r r r

φ φ φ φμφ φ φ

⎡ ⎤∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞Φ = + + + + + + + + + −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

1 (a) Starting with Cauchy’s equations of motion, j ii

i

j

Dvg

Dt x

τρ ρ

∂= +

∂, derive an equation representing

local balance of mechanical energy. Identify the terms in your equation that represent rate of change

of kinetic and potential energy. Here, ρ is the density of the fluid, iiDx

vDt

= is the component of the

fluid velocity, v, in the ix -direction, ig is the component of the gravitational acceleration, g , in the

direction of the coordinate axis ix , i jτ is the stress tensor, t is the time and D

Dt denotes material

derivative.

(b) Consider a material volume of fluid (closed system) that is translating, rotating and deforming in a

gravitational force field. The net rate at which work is done on the system by surface forces exerted by the

surrounding fluid is ( , , )

m m

s i i i ji j

B B

W S t v dA v n dAτ= =∫ ∫n r , where mB is the surface bounding the

material region mR , 1 2 3x x x= + +1 2 3r e e e denotes the position vector of a point, ie is the unit vector in the ix -direction, n is the unit outward normal at a point on the material surface mB and ( , , )iS tn r is the

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stress vector at a point on a plane normal to n . Using the Gauss divergence theorem and your answer to

part (a), show that sW may be expressed as

m

s s

R

W w dV= ∫ , where sw is the sum of four groups of terms:

1( ) ( ) . ,2

s ji ij

Term I Term II Term III Term IV

D Dw p d

Dt Dtρ ρ π= ⋅ + − ⋅ − ∇ +v v g r v

p is the thermodynamic pressure, ijπ is the deviatoric stress tensor and ijd is the deformation tensor or rate of strain tensor.

(c) Identify the term or terms (I to IV) in your answer to part (b) that result in a change in the internal

energy of the system. Give reasons to justify your answer.

(d) Identify the term or terms (I to IV) in your answer to part (b) that lead to a generation in entropy.

(e) Identify the terms (I to IV) in your answer to part (b) that represent reversible modes of work and the

terms that represent irreversible modes of work. Give reasons to justify your answer.

[4+8+2+2+4=20 marks]

2. The energy equation may be expressed in ‘ vc form’ as

2

v

T

DT Tc k T q

Dt

βρκ

′′′= ∇ + +Φ − ∇⋅ v , (I)

where T is the temperature, vc is the specific heat capacity of the fluid at constant volume, k is the

thermal conductivity of the fluid, Φ is the viscous dissipation, q′′′ is the local rate of volumetric heat generation,β is the coefficient of thermal expansion of the fluid and Tκ is the isothermal compressibility of the fluid, or in ‘

pc form’ as

2

p

DT Dpc k T q T

Dt Dtρ β′′′= ∇ + +Φ + , (II)

where pc is the specific heat capacity of the fluid at constant pressure. Consider low Mach number flow

of a calorically perfect gas. It was shown in the class that the viscous dissipation term and the term

involving Dp

Dt may be neglected in comparison with the other terms in equation (II) when the Mach

number is small, so that equation (II) may be approximated as

2

p

DTc k T q

Dtρ ′′′= ∇ + . (III)

For low Mach number flows of a gas, the continuity equation may be approximated by

0∇⋅ =v . (IV) The relation (IV) suggests that equation (I) may be approximated, for low Mach number flows, by

2

v

DTc k T q

Dtρ ′′′= ∇ + . (V)

Since pc and vc for gases are not equal, the two approximate equations, (III) and (V), are not identical.

Which of the two equations, equation (III) or equation (V), represents the correct approximation ? Give

reasons to justify your answer. Suggest an explanation for this ‘energy equation paradox’. [5 marks]

3. Consider plane Couette flow of a constant-property Newtonian-Fourier fluid between two large

parallel horizontal plates that are separated by a distance L. The top plate moves in its own plane, with

speed U . The bottom plate is stationary. There is no externally applied pressure gradient. The velocity

profile is given by ( / )u U y L= , where u is the component of velocity in the x-direction and y is the coordinate normal to the walls, measured from the bottom plate. The streamlines are parallel to the x-

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axis. The fluid liberates thermal energy uniformly at a constant rate, S, per unit volume. The bottom plate

is maintained at a constant temperature 1T . The top plate is maintained at a constant temperature 2T .

(a) Starting with an appropriate form of the thermal energy equation, derive an ordinary differential

equation for the temperature distribution ( ).T y Include the effect of viscous dissipation and internal heat

generation in your analysis.

(b) Express the equation derived in part (a) in non-dimensional form, using the following non-

dimensional variables: 1

2 1

T T

T Tθ −=

−,

yY

L= . Identify the non-dimensional parameters governing the

non-dimensional temperature distribution ( )Yθ . (c) Obtain appropriate boundary conditions for the non-dimensional equation of part (b).

(d) Determine the non-dimensional temperature distribution ( )Yθ .

(e) Determine the Nusselt number, hL

Nuk

= , at the bottom plate, where h is the heat transfer coefficient

based on the temperature difference between the two plates and k is the thermal conductivity of the fluid.

[4+2+2+4+3=15 marks]

4. Consider the flow of a constant-property Newtonian-Fourier fluid through a circular duct of inside

radius 0r , surrounded by air at a constant ambient temperature T∞ . The axial velocity distribution in the

hydrodynamically developed region is given by

2

2

0

( ) 2 (1 )z avr

v w r wr

= = − , where r is the radial

coordinate and avw is the average velocity. The radial component of velocity, rv , and the swirl or

azimuthal component of velocity, vφ , are zero. Assume that the temperature distribution is a function of

the radial coordinate only. The effective external heat transfer coefficient, eh , may be assumed to be

constant. The walls of the duct may be assumed to be perfectly conducting.


equation for the temperature distribution ( )T r . Include the effect of viscous dissipation in your analysis.

(b) Write appropriate boundary conditions for solving the ordinary differential equation of part (a).

(c) Determine the radial distribution of temperature, ( )T r , in the fluid.

(d) Determine the wall temperature, wT , using your answer to part (c).

(e) Determine the centre-line temperature, 0T , using your answer to part (c).

(f) Determine the limiting form of the temperature distribution when the Biot number, 0(2 )eh rBik

= , is

very large, that is, 1Bi >> . Here, k is the thermal conductivity of the fluid flowing through the duct. (g) What physical situation does the limiting case, 1Bi >> , or Bi →∞ , represent ? (h) What physical situation does the limiting case, 1Bi



No. of Students: 47 End Spring Semester Examination 2012




wherever necessary. Please state your assumptions clearly. The following information may be useful:

1

0

0,

sin( )sin( ) 1,

2

m n

m Y n Y dYm n

π π≠⎧

⎪= ⎨=⎪⎩

∫ , where m and n are integers.

1

0

66 (1 ) , 2

( 2)( 3)

m

mI d when mm m

η η η η= − = > −+ +∫ .

1. Consider steady laminar fully developed forced-convection flow of a constant-property Newtonian

Fourier fluid between two large parallel plates, driven by a constant applied pressure gradient in the x-

direction. There is a constant heat flux, 0q , into the fluid from the bottom plate. The top plate is insulated.

The fully developed velocity distribution is given by ( )6 1avu u η η= − , where u is the component of

velocity in the x-direction, avu is the average velocity through the channel, y

Lη = , L is the distance

between the plates and y is the coordinate normal to the plates, measured from the bottom plate. The

plates have infinite span (in the z-direction). The effect of viscous dissipation and axial conduction in the

fluid are negligible.

(a) Show that the bulk temperature of the fluid is given by

1

0

6 (1 )bT T dη η η= −∫ , where T is the temperature of the fluid.

(b) Obtain an expression for the bulk temperature gradient, bdT

dx, by integrating the energy equation across

the channel.

(c) In the thermally developed region of the flow, w

b w

T T

T T

−−

is a sole function of the transverse coordinate,

η . Using this relation, show that the local heat transfer coefficient at the bottom plate, 0w b

qh

T T=

−, is a

constant in the thermally developed region. Here, ( )wT x is the temperature of the bottom plate.

(d) Using the result of part (c), show that (i) w bdT dT

dx dx= and (ii) b

T dT

x dx

∂=

∂ in the thermally developed

region.

(e) Determine the temperature distribution in the fluid relative to the temperature of the bottom plate, in

the thermally developed region of the flow.

(f) Determine the Nusselt number at the bottom plate. [2+2+2+4+5+5=20 marks]

2. Consider steady laminar two-dimensional incompressible forced-convection boundary-layer flow of a

constant-property Newtonian fluid over a semi-infinite flat plate aligned with the direction of a uniform

isothermal oncoming free-stream. The plate is maintained at a constant temperature, wT , higher than the

temperature, T∞ , of the free-stream. The boundary-layer equations admit a similarity solution of the form

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( )u

u Fu

η∞

′= = , [ ]1/ 2 1Re ( ) ( )2

vV F F

u xη η η

∞

′= = − , ( )ww

T TG

T Tθ η

∞

−= =

−,

where ( )F η is the solution of the Blasius equation for flat plate flow, 0F FF′′′ ′′+ = , with boundary conditions (0) (0) 0F F′= = , ( ) 1F ′ ∞ = , and ( )G η is the solution of the equation, Pr ( ) 0G F Gη′′ ′+ = ,

with boundary conditions (0) 0G = and ( ) 1G ∞ = . Here, 2

Y

xη = , xx

L= , 1/ 2ReY y= , Re

u L

ν∞= ,

yy

L= , u is the component of fluid velocity in the x -direction, v is the component of fluid velocity in

the y -direction, T is the fluid temperature, x is the coordinate along the plate, y is the coordinate

normal to the plate, measured from the plate, u∞ is the free-stream speed, L is a reference length, ν is the kinematic viscosity of the fluid and Pr is the Prandtl number of the fluid. The values of ,F F ′ and F ′′ at selected values of η are given in the table below.

η ( )F η ( )F η′ ( )F η′′ 0 0 0 0.46960

0.1 0.00235 0.04696 0.46956

0.2 0.00939 0.09391 0.46931

3.4 2.18747 0.98797 0.03054

3.6 2.38559 0.99289 0.01933

(a) Obtain a relation between the local Nusselt number, xNu , and the local Reynolds number, Rex .

(b) The non-dimensional thickness, TTL

δδ = , of the thermal boundary layer varies with x as

1/ 2Re mT a xδ−= , where a and m are constants, and Tδ is the dimensional thermal boundary-layer

thickness. Use the similarity solution to determine the value of m.

(c) Use your answer to part (a) and the above table to determine the value of 1/ 2Rex xNu−

when Pr = 1.

Give a rigorous mathematical justification for your answer.

(d) The value of the constant, a, in part (b) depends on the criterion used to define the thickness of the

thermal boundary layer. Using the 99% criterion and the above table, estimate the value of a when Pr = 1.

Give reasons to justify your answer. [5+3+4+3 =15marks]


large Prandtl number constant-property Newtonian fluid over a semi-infinite flat plate aligned with the

direction of a uniform isothermal oncoming free-stream. The plate is maintained at a constant temperature,

wT , higher than the temperature, T∞ , of the free-stream.

(a) Estimate the order of magnitude of the non-dimensional velocity, u

uu∞

= , inside the thermal boundary

layer. Express your answer in terms of the non-dimensional thickness, TTL

δδ = , of the thermal boundary

layer and the non-dimensional thickness, L

δδ = , of the hydrodynamic boundary layer. Here, u is the

component of fluid velocity parallel to the plate, u∞ is the free-stream speed, δ is the dimensional thickness of the hydrodynamic boundary layer, Tδ is the dimensional thickness of the thermal boundary layer, and L is a reference length.

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(b) The ratio of the thicknesses of the thermal and hydrodynamic boundary layers, Tδδ

, depends on the

Prandtl number and is given by 1

Pr

T

m

δδ

= in the limit as Pr →∞ , where m is a constant and Pr is the

Prandtl number of the fluid. Use scaling arguments to determine the value of m.

(c) The local Nusselt number at distance L from the leading edge varies as ~ Re Prn qLNu when Pr 1>> ,

where n and q are constants, Reu L

ν∞= and ν is the kinematic viscosity of the fluid. Use scaling

arguments to determine the values of n and q. [2+4+4=10 marks]

4. Consider steady laminar two-dimensional natural-convection boundary-layer flow of a Newtonian fluid

along a semi-infinite heated vertical flat plate kept at a constant temperature, wT , higher than the ambient

temperature, T∞ . The boundary-layer equations admit a similarity solution of the form

( )1/ 20

4 ( )u

u x Fu

η′= = , ( )

[ ]1/ 4 1/ 40

1( ) 3 ( )

4

vV Gr F F

u xη η η′= = − , ( )

w

T TG

T Tθ η∞

∞

−= =

−,

where xxL

= , 0 ( )wu g T T Lβ∞ ∞= − , 3 2( ) /wGr g T T Lβ ν∞ ∞= − , ( )F η and ( )G η are solutions of the

equations 23 2( ) 0F FF F G′′′ ′′ ′+ − + = , 3Pr ( ) 0G F Gη′′ ′+ = , subject to the boundary conditions

(0) (0) 0F F′= = , (0) 1G = , ( ) 0F ′ ∞ = , ( ) 0G ∞ = , 1/ 4(4 )

Y

xη = , 1/ 4Y y Gr= , yy

L= g is the

gravitational acceleration, β∞ is the coefficient of thermal expansion at the ambient temperature, ν is the kinematic viscosity of the fluid, u is the component of fluid velocity in the x -direction, v is the component of fluid velocity in the y -direction, T is the fluid temperature, x is the coordinate along the

plate, pointing vertically upwards, y is the coordinate normal to the plate, measured from the plate, and L

is a reference length. The following table gives the values of F, G and their derivatives for Pr = 1, at

selected values of η :

η ( )F η ( )F η′ ( )F η′′ ( )G η ( )G η′ 0 0 0 0.6421 1 -0.5671

0.1 0.0030 0.0593 0.5450 0.9433 -0.5669

6.0 0.5194 0.0004 -0.0014 0.0002 -0.0005

6.25 0.5194 0.0000 -0.0010 0.0000 -0.0004

(a) Show that 1/ 4

x xNu c Gr= , where xNu is the local Nusselt number, xGr is the local Grashof number and c is a constant that depends on the Prandtl number.

(b) Use the relation between the local Nusselt number and the local Grashof number given in part (a) to

obtain an expression for the total heat transfer rate, Q, over a length L of the plate from the leading edge,

per unit width in the spanwise direction (z-direction). Express your answer in terms of the constant c

defined in part (a).

(c) Show that the transverse velocity at the edge of the boundary layer is 1/ 4 1/ 4

0v u Gr b x− −

∞ = , where b is

a constant that depends on the Prandtl number. Use the fact that ( ) 0F η′ → at a rate faster than 1η − as η →∞ . (d) Use the above table to determine the values of b and c when Pr = 1.

(e) Consider natural convection along a semi-infinite porous plate, with a suction velocity,

( ,0) mV x Kx= − applied at the wall, where K and m are constants, and K is positive.

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(i) Determine the value of m for which a similarity solution of the form indicated above exists.

(ii) Modify the boundary conditions on F and G to account for the effect of suction at the porous plate

when m takes the value determined in part (i). [5+6+3+2+4=20 marks]

5. A low Prandtl number constant-property Newtonian fluid enters the gap between two large heated

parallel plates. The distance between the plates is L. Both the plates are maintained at constant

temperature wT . The temperature of the fluid at the inlet is iT . The speed of the fluid at the inlet is U .

Here, wT , iT and U are constants, and w iT T≠ . For low Prandtl number fluids, the temperature profile in a duct develops more rapidly than the velocity profile. In such a situation, the temperature profile can be

determined based on a uniform velocity profile. This is called the “slug flow” solution. Assume that the

velocity distribution in the channel can be approximated by u U= , where u is the component of velocity in the direction parallel to the plates. The effect of viscous dissipation and axial conduction in the fluid

may be neglected.

(a) Write the energy equation describing the temperature distribution in the fluid for the physical situation

described above, in Cartesian coordinates (x, y), where x is the coordinate along the plates and y is the

coordinate normal to the plates.

(b) Non-dimensionalize the equation of part (a) using the following non-dimensional variables:

w

i w

T T

T Tθ −=

−,

yY

L= ,

xX

L Pe= , where

ULPe

α= and α is the thermal diffusivity of the fluid.

(c) Write appropriate boundary conditions for solving the partial differential equation of part (b).

(d) Use the method of separation of variables to obtain the solution, ( , )X Yθ , of the partial differential equation derived in part (b), subject to the boundary conditions of part (c).

(e) Use the series solution of part (d) to obtain an expression for the non-dimensional bulk temperature,

b wb

i w

T T

T Tθ −=

−. Express your answer in terms of the coefficients of the series solution of part (d).

(f) Obtain an expression for the variation of the local Nusselt number at the bottom plate. Express your

answer in terms of the coefficients of the series solution of part (d).

(g) Determine the value of the Nusselt number in the thermally developed region, using your answer to

part (f). [2+2+3+8+2+4+4 = 25 marks]

6. Give concise answers to the following parts, with proper justification.

(a) Consider hydrodynamically developed and thermally developing forced convection in a circular tube

with step change in the wall temperature. A student argues that the thermal entrance length is larger in the

case of forced convection of water than in the case of forced convection of air when the velocity

distribution is identical for the two cases. Do you agree with the student ?

(b) Consider forced convection in a channel. A student argues that the effect of viscous dissipation is more

important in the case of forced convection of oil than in the case of forced convection of liquid mercury

when the velocity distribution is identical for the two cases. Do you agree with the student ?

(c) Consider forced convection from a large heated isothermal plate aligned with the direction of a

uniform isothermal oncoming free-stream of air at low Mach numbers. A student argues that the heat

transfer from the plate will be doubled if the free-stream speed is doubled, for the same temperature

difference between the plate and the free-stream. Do you agree with the student ?

(d) Consider forced convection from a large heated isothermal plate aligned with the direction of a

uniform isothermal oncoming free-stream of air at low Mach numbers. A student argues that the heat

transfer from the plate will be doubled if the temperature difference between the plate and the free-stream

is doubled and the free-stream speed is kept the same. Do you agree with the student ?

(e) Consider natural convection heat transfer to air from a large heated vertical isothermal plate. A student

argues that the heat transfer from the plate will be doubled if the difference between the plate temperature

and the ambient temperature of the surrounding air is doubled. Do you agree with the student ?

[2+2+2+2+2=10 marks]

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No. of Students: 52 Mid Spring Semester Examination 2013




wherever necessary. Please state your assumptions clearly.

The following information may be useful:

The divergence theorem states that j

j j

jB R

aa n d A dV

x

∂=


p

p

cs

T T

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠, v

v

s c

T T

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠,

T v

s p

v T

∂ ∂⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠,

pT

s v

p T

⎛ ⎞∂ ∂⎛ ⎞= −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠,

v T

p

T

βκ

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠ where

1

p

v

v Tβ ∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠

and1 1

T

T T

v

p v p

ρκρ⎛ ⎞ ⎛ ⎞∂ ∂

= = −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

1 (a) By integrating Cauchy’s equations of motion,j ii

i

j

Dvg

Dt x

τρ ρ

∂= +

∂, over a material region, ( )mR t ,

bounded by a material surface, ( )mS t , and using the relation,( ) ( )m mR t R t

d DdV dV

dt Dt

φρφ ρ=∫∫∫ ∫∫∫ , show that the time rate of change of mechanical energy (sum of kinetic energy and potential energy) of a material

region (closed system) occupied by a specified portion of fluid (material) that is translating, rotating and

deforming in a gravitational force field is given by

( ) ( ) ( ) ( )

1.

2

1 2 3

m m m m

i ji j

R t S t R t R t

ddV v n dA p dV dV

dt

Term Term Term

ρ τ⎛ ⎞⋅ − ⋅ = + ∇⋅ − Φ⎜ ⎟⎝ ⎠∫∫∫ ∫∫ ∫∫∫ ∫∫∫v v g r v

Here, ρ is the density of the fluid, iiDx

vDt

= is the component of the fluid velocity, v, in the ix -

direction, ig is the component of the gravitational acceleration, g , in the direction of the coordinate axis

ix , t is the time,D

Dt denotes material derivative, r is the position vector,

jn is the jth component of the

unit outward normal, n̂ , at a point on the material surface, p is the thermodynamic pressure, ji ijdπΦ =

is the viscous dissipation function, and i jτ , ijπ and 1

2

jiij

j i

vvd

x x

⎛ ⎞∂∂= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

are respectively the elements of

the stress tensor, the deviatoric stress tensor, and the deformation tensor or the rate of strain tensor.

(b) By integrating the ‘internal energy form’ of the equation representing local balance of thermal

energy, De

p qDt

ρ ′′′= −∇⋅ − ∇⋅ +Φ +q v , over a material region, show that the time rate of change of

internal energy of a material region of fluid that is translating, rotating and deforming is given by

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( ) ( ) ( ) ( ) ( )

ˆ .

4 5 6 7

m m m m mR t S t R t R t R t

dedV dA p dV dV q dV

dt

Term Term Term Term

ρ ′′′= − ⋅ − ∇⋅ + Φ +∫∫∫ ∫∫ ∫∫∫ ∫∫∫ ∫∫∫q n v

Here, e is the specific internal energy, q is the heat flux vector and q′′′ is the local rate of volumetric heat generation.

(c) Using your answers to parts (a) and (b), obtain an equation for the time rate of change of total energy

(sum of mechanical energy and internal energy) of a material region of fluid that is translating, rotating

and deforming in a gravitational force field.

(d) Identify the term or terms (Term1 to Term 7) that result in a change in the total energy of the system.

Give reasons to justify your answer.

(e) Identify the term or terms (Term 1 to Term 7) that represent irreversible conversion of mechanical

energy to internal energy. Give reasons to justify your answer.

(f) Identify the term or terms (Term 1 to Term 7) that represent reversible conversion of mechanical

energy to internal energy or vice versa. Give reasons to justify your answer.

[10+2+2+2+2+2=20 marks]

2. Consider steady forced convection of a constant-property Newtonian-Fourier fluid in the gap between

two large parallel horizontal plates that are separated by a distance H. The top plate moves in its own

plane, in the x-direction, with constant speed U . The bottom plate is stationary. There is no externally

applied pressure gradient. The velocity field is given by ( / )u U y H= , 0v = , 0w = , where u, v and w are the components of velocity in the x, y and z directions respectively, and y is the coordinate normal to

the walls, measured from the top surface of the bottom plate. The streamlines are parallel to the x-axis.

The bottom plate is uniformly heated, with a constant heat flux 1q . Heat transfer from the top plate to the

surrounding air which is at a constant ambient temperature T∞ , may be represented by a convective

boundary condition, with constant external heat transfer coefficient eh . The thermal conductivity, k, of

the fluid and the thermal conductivity, wk , of the walls of the channel are constant.


equation for the temperature distribution, ( )T y , in the fluid. Include the effect of viscous dissipation in

your analysis.

(b) Starting with the equation for heat conduction in a solid, show that the transverse gradient, sdT

dy, of

the temperature in the top plate is constant. Here, ( )sT y is the temperature distribution in the top plate.

(c) Express the equation derived in part (a) in non-dimensional form, using the following non-

dimensional variables: 1

T T

q H

k

θ ∞−= , yYH

= .

(d) Obtain an appropriate non-dimensional boundary condition for the non-dimensional equation of part

(c), at the interface between the fluid and the top surface of the bottom plate ( 0Y = ).

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(e) Using your answer to part (b) or otherwise, show that the boundary condition for the non-dimensional

equation of part (c), at the interface between the fluid and the bottom surface of the top plate, may be

expressed in non-dimensional form as

0d

BidY

θ θ+ = at 1Y = ,

where Bi is an effective Biot number defined by effh H

Bik

= , that may be determined using the relation

1 1 w

e w

k t

Bi Bi k H= + ,

effh is an effective heat transfer coefficient, eBi is the Biot number based on the external heat transfer

coefficient, given by eeh H

Bik

= , and wt is the thickness of the top plate.

(f) Without solving the equation obtained in part (c), show that the non-dimensional temperature

distribution depends on the Biot number, Bi , and the modified Brinkman number, Br , defined by * PrBr Ec= , using your answers to parts (c), (d) and (e), where *Ec is a modified Eckert number defined

by

2*

1p

UEc

q Hc

k

= , pc is the specific heat capacity of the fluid and Pr is the Prandtl number of the fluid.

(g) Determine an approximate form of the non-dimensional equation obtained in part (c) when the effect

of viscous dissipation is negligible. Solve this approximate equation to obtain the distribution of the non-

dimensional temperature,θ , when the effect of viscous dissipation is negligible.

(h) Without solving the equation obtained in part (c), show that the solution of the non-dimensional

equation of part (c) may be expressed as

1 2( ; , ) ( ; ) ( ; )Y Bi Br Y Bi Br Y Biθ θ θ= + , where 1( ; )Y Biθ is the temperature distribution obtained in part (g) when the effects of viscous dissipation is negligible, using the principle of superposition. Obtain a differential equation for 2( ; )Y Biθ , and appropriate boundary conditions for this differential equation.

(i) Determine the function, 2( ; )Y Biθ , by solving the differential equation derived in part (h), subject to appropriate boundary conditions.

(j) Using your answers to parts (g), (h) and (i), obtain an expression for the temperature of the lower

surface of the top plate.

(k) Using your answers to parts (g), (h) and (i), determine the temperature distribution in the fluid for the

special case when the inside surface of the upper plate, which is adjacent to the fluid, is at temperature

T∞ . Do not solve the equation with the boundary condition T T∞= at y = H, to answer this question.

(l) Determine the limiting form of the temperature distribution when the Biot number is very small, that

is, 1Bi

3 (a) The local balance of thermal energy may be expressed in ‘entropy form’ as

DsT q

Dtρ ′′′= −∇ ⋅ + +Φq ,

where ρ is the density of the fluid, T is the temperature, s is the specific entropy, t is the time, DDt

denotes material derivative, q is the heat flux vector, q′′′ is the local rate of volumetric heat generation and Φ is the viscous dissipation. Starting with the above equation, use appropriate thermodynamic relations to show that the energy equation for a ‘Fourier’ fluid may be expressed in ‘ vc form’ as

( )vT

DT Tc k T q

Dt

βρκ

′′′= ∇ ⋅ ∇ + +Φ − ∇⋅ v ,

where vc is the specific heat capacity of the fluid at constant volume, k is the thermal conductivity of the

fluid, β is the coefficient of thermal expansion of the fluid, Tκ is the isothermal compressibility of the fluid and v is the fluid velocity.

(b) Two students, A and B, were asked to obtain an approximate form of the energy equation for low

Mach number flow of a gas. Both the students, A and B, agree that the viscous dissipation term may be

neglected since the Eckert number is small for low Mach number flows. Student A argues that the last

term on the right hand side of the ‘ vc form’ of the energy equation stated in part (a), involving ∇⋅ v , may be neglected since the continuity equation is approximated by 0∇⋅ =v for low Mach number flows of a gas, and hence, the equation may be approximated as

( )vDT

c k T qDt

ρ ′′′= ∇ ⋅ ∇ + .

Student B thinks that this term, involving ∇⋅ v , cannot be neglected even if the Mach number is small. Which student, A or B, do you think is correct ? Give reasons to justify your answer.

[5+5=10 marks]

.

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Sub. No.: ME60014 Sub. Name: Convective Heat and Mass Transfer

Instructions: Attempt all questions. This question paper consists of four pages. Symbols have their usual

meanings. Please explain your work carefully. Make suitable assumptions wherever necessary. Please

state your assumptions clearly. The following information may be useful:

2

02

te dtπ∞ − =∫ ,

3

0

0.893te dt

∞− =∫ ,

21( ) ( ) constantzerfc z d z z erfc z e

π−= − +∫ .

lim ( ) 0erfcη

η η→∞

=

1. Consider steady laminar forced-convection flow of a low Prandtl number constant-property Newtonian

Fourier fluid between two large parallel plates, driven by an applied pressure gradient in the x-direction.

The distance between the plates is L. There is a constant heat flux, 0q , into the fluid from the bottom plate.

The top plate is insulated. The velocity profile at the entrance of the parallel-plate channel is uniform. The

speed of the fluid at the inlet of the channel is U . For low Prandtl number fluids, the temperature profile in a duct develops more rapidly than the velocity profile. In such a situation, the temperature profile can

be obtained based on a uniform velocity profile. This is called the ‘slug flow’ solution. Assume that the

velocity distribution in the channel can be approximated by u U= , where u is the component of velocity in the x-direction. The plates have infinite span (in the z-direction). The effect of viscous dissipation and

axial conduction in the fluid may be neglected.

(a) Show that the bulk temperature of the fluid is given by

1

0

bT T dη= ∫ , where T is the temperature of the

fluid, y

Lη = , y is the coordinate normal to the plates, measured from the bottom plate.

(b) Obtain an expression for the bulk temperature gradient, bdT

dx, by integrating the energy equation across

the channel.

(c) In the thermally developed region of the flow, w

b w

T T

T T

−−


η . Using this relation, show that the local heat transfer coefficient at the bottom plate, 0w b

qh

T T=

−, is a

constant in the thermally developed region. Here, ( )wT x is the temperature of the bottom plate.

(d) Using the result of part (c), show that (i) w bdT dT

dx dx= and (ii) b

T dT

x dx

∂=


region.

(e) Determine the temperature distribution in the fluid relative to the temperature of the bottom plate, in

the thermally developed region of the flow.

(f) Determine the Nusselt number at the bottom plate. [2+2+2+4+5+5=20 marks]

2. Consider large Peclet number steady laminar axisymmetric hydrodynamically developed

incompressible flow of a constant-property Newtonian Fourier fluid through a circular duct of radius 0r ,

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2

with step change in wall temperature, given by 0, 0

( , ), 0

i

w

T zT r z

T z

3

Here the bars denote dimensional quantities, L is a reference length, 0u is a reference velocity,

wT T T∞Δ = − , x is the vertical coordinate, measured from the leading edge of the plate, y is the horizontal coordinate normal to the plate, measured from the plate, u and v are the components of velocity in the x

and y directions respectively, T is the temperature, p is the pressure, ρ∞ is the density of the fluid at temperature T∞ , and g is the gravitational acceleration.

(b) Determine the reference velocity, 0u , by equating the order of magnitudes of the inertia and buoyancy

force terms in the vertical momentum equation.

(c) Use scale analysis to estimate

(i) the order of magnitude, 0δ , of the non-dimensional thickness of the boundary layer, and

(ii) the order of magnitude, 0v , of the non-dimensional horizontal velocity, v, inside the boundary layer.

Express your answer in terms of the Grashof number3 2/Gr g T Lβ υ∞= Δ , where β∞ is the coefficient of

thermal expansion at the ambient temperature and υ is the kinematic viscosity of the fluid. (d) Express the system of equations of part (a) using scaled variables, 0 /Y y δ= , 0 / V v v= . Obtain

the boundary-layer equations that describe the flow for large values of Gr .

(e) Write appropriate boundary conditions for the system of equations of part (d).

(f) The boundary-layer equations for natural convection along a semi-infinite flat plate admit a similarity

solution of the form ( )ax Fψ η= , = ( )bx Gθ η , satisfying the boundary conditions of part (e), where

ψ is the streamfunction defined by the equations uY

ψ∂=∂

, Vx

ψ∂= −

∂, and

m

Y

xη = . Determine the

values of a, b, and m.

(g) Use your answer to part (f) to show that n

x xNu c Gr= , where xNu is the local Nusselt number, xGr is the local Grashof number and c is a constant that depends on the Prandtl number. Determine the value of

n. [4+2+4+4+3+5+3=25 marks]


low Prandtl number constant-property Newtonian Fourier fluid over a semi-infinite flat plate aligned

with the direction of a uniform isothermal oncoming flow with flow speed u∞ and temperature T∞ . The

plate is subjected to a constant surface heat flux, 0q . The velocity field may be approximated by u u∞= , 0v = in the entire thermal boundary layer. Here, u and v are the components of the velocity in the x and

y directions, x is a coordinate along the plate, measured from the leading edge, y is the coordinate normal

to the plate, measured from the plate.

(a) Write the boundary-layer energy equation using the approximate velocity field given above, in

dimensional form.

(b) By differentiating the boundary-layer energy equation show that the heat flux in the direction normal

to the plate, yT

q ky

∂= −

∂, satisfies the same differential equation as the temperature distribution.

(c) Obtain suitable boundary conditions for the partial differential equation of part (b).

(d) Show that the solution of the partial differential equation of part (b), subject to the boundary

conditions obtained in part (c), is given by

( )0yq q erfc η= ,

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where

2

y

x

u

ηα

∞

= , α is the thermal diffusivity of the fluid, ( )erfc η is the complementary error

function defined by ( ) 1 ( )erfc erfη η= − , and ( )erf η is the error function defined by 2

0

2( ) terf e dt

η

ηπ

−= ∫ . (e) Use your answer to part (d) to determine the temperature distribution ( , )T x y .

(f) Determine the variation of the wall temperature, ( )wT x , with distance along the plate.

(g) Obtain an expression for the local Nusselt number. Express your answer in terms of the local

Reynolds number and the Prandtl number. [1+2+2+10+4+2+4=25 marks]


(a) Convective heating or cooling may be described under certain circumstances by Newton’s law of

cooling, which states that the rate of heat loss of a body is directly proportional to the difference in

temperatures between the body and its surroundings. Does Newton’s law of cooling hold for natural

convection from a large vertical isothermal plate ?

(b) Consider forced convection from a large heated isothermal plate aligned with the direction of a

uniform isothermal oncoming free-stream of air at low Mach numbers. An engineer wants to double the

rate of heat transfer from the plate by increasing the free-stream speed. By what factor should he increase

the free-stream speed in order to increase the rate of heat transfer from the plate by a factor of two ?

(c) Consider steady laminar axisymmetric hydrodynamically developed and thermally developing forced

convection of water in a circular tube with step change in the wall temperature. The Reynolds number

based on the average velocity through the duct and the diameter of the tube is 500. A student argues that

the thermal entrance length will be halved if the volume flow rate is doubled. Do you agree with the

student ?

(d) Consider steady laminar axisymmetric hydrodynamically and thermally developed forced convection

of water in a circular tube. The inside surface of the tube is at a constant temperature higher than the

temperature of the fluid at the inlet. An engineer thinks that the heat flux at the surface of the tube in the

thermally developed region of the flow will be doubled if the flow rate is doubled. Do you agree with the

engineer ?

(e) Consider forced convection in a channel. A student argues that the effect of viscous dissipation is

more important in the case of forced convection of water than in the case of forced convection of oil when

the velocity distribution is identical for the two cases. Do you agree with the student ?

[2+2+2+2+2=10 marks]

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wherever necessary. Please state your assumptions clearly. The following information may be useful:


j j

jB R

aa n d A dV

x

∂=


( ) ( )m mR t R t

D DdV dV

Dt Dt

φρ φ ρ=∫ ∫

1. (a) Show that the rate at which work is done by surface forces acting on a material region, mR (t), of

fluid (closed system) that is translating, rotating and deforming in a gravitational force field may be

expressed as

( )ms s

R t

W w dV= ∫& & , where sw& is the sum of four groups of terms:

1( ) ( ) . ,2

s

Term I Term II Term III Term IV

D Dw p

Dt Dtρ ρ= ⋅ + − ⋅ − ∇ + Φ& v v g r v

ji ijdπΦ = , ijπ is the deviatoric stress tensor, ijd is the deformation tensor or rate of strain tensor, ρ is the density of the fluid, v is the fluid velocity, g is the gravitational acceleration, p is the thermodynamic

pressure, t is the time, D

Dt denotes material derivative and r denotes position vector.

(b) Using your answer to part (a), show that De

pDt

ρ = −∇⋅ − ∇ ⋅ +Φq v , where e is the specific internal

energy and q is the heat flux vector.

(c) Using your answer to part (b), show that Ds

TDt

ρ = −∇⋅ +Φq , where T is the absolute temperature

and s is the specific entropy.

(d) Using your answer to part (c), show that the rate of production or generation of entropy may be

expressed as

( )ms s

R t

P p dV= ∫& & , where

2s

Term A Term B

Tp .

T T

Φ ⋅∇= −

q&

.

(e) Obtain sufficient conditions on Term A ( T

Φ )and Term B (

2

T

T

⋅∇q ) in your answer to part (d) such

that the Clausius-Duhem inequality, 0sp ≥& , holds for all possible circumstances. (f) Identify the term or terms (I to IV) in your answer to part (a) that result in generation of entropy. Give

reasons to justify your answer.

(g) Identify the terms (I to IV) in your answer to part (a) that represent reversible modes of work and the

terms that represent irreversible modes of work. Give reasons to justify your answer.

[10+4+4+4+2+2+4=30 marks]

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2. Consider steady laminar forced convection of a constant-property Newtonian-Fourier fluid in the gap

between two large parallel horizontal plates that are separated by a distance L, with a constant applied

pressure gradient in the x-direction. The bottom plate is maintained at a constant temperature 1T . The top

plate is uniformly heated, with a constant heat flux, 2q , into the fluid. The fully-developed velocity field

is given by 6 ( / )(1 / )avu u y L y L= − , 0v = , 0w = , where u, v and w are the components of velocity in the x, y and z directions respectively, avu is the average velocity through the channel and y is the

coordinate normal to the walls, measured from the bottom plate. Assume that the plates have infinite span

in the z-direction. Considering the effect of viscous dissipation, determine

(a) the temperature distribution in the fluid,

(b) the temperature of the top plate, and

(c) the Nusselt number, hL

Nuk

= , at the bottom plate, where h is the heat transfer coefficient based on

the temperature difference between the two plates and k is the thermal conductivity of the fluid.

[10+1+4=15 marks]

3. Consider steady laminar forced-convective heat transfer to a low Prandtl number constant-property

Newtonian-Fourier fluid in a circular duct of radius, R, subjected to uniform surface heat flux wq . The

velocity profile at the entrance of the duct is uniform. The speed of the fluid at the inlet of the duct is 0W .

For low Prandtl number fluids (e.g. liquid metals), the temperature profile in a duct develops more rapidly

than the velocity profile. In such a situation, the temperature distribution can be obtained based on a

uniform velocity profile. This is called the ‘slug flow’ solution. Accordingly, consider thermally

developed flow with a velocity distribution approximated by 0w W= , where w is the axial component of the fluid velocity. Neglecting the effect of viscous dissipation and axial conduction in the fluid, determine

(a) the variation of the bulk temperature, bT , of the fluid in the axial direction,

(b) the variation of the temperature, wT , of the duct wall in the axial direction in the thermally developed

region of the flow,

(c) the temperature distribution in the fluid, relative to the temperature, wT , of the duct wall, in the

thermally developed region of the flow, and

(d) the Nusselt number based on the duct diameter.

[3+3+5+4=15 marks]

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state your assumptions clearly. The following information may be useful: 3

0

0.892979512e dξ ξ∞

− =∫

1. Consider steady laminar flow of a low Prandtl number constant-property Newtonian Fourier fluid

between two large parallel plates, driven by an applied pressure gradient in the x-direction. The distance

between the plates is L. The bottom plate is insulated. There is a constant heat flux, 0q , into the fluid from

the top plate. The fluid flowing through the channel is heated when 0 0q > and cooled when 0 0q < . The velocity profile at the entrance of the parallel-plate channel is uniform. The speed of the fluid at the inlet

of the channel is U . For low Prandtl number fluids, the temperature profile in a duct develops more rapidly than the velocity profile. In such a situation, the temperature profile can be obtained based on a

uniform velocity profile. This is called the ‘slug flow’ solution. Assume that the velocity distribution in

the channel can be approximated by u U= , where u is the component of velocity in the x-direction. The plates have infinite span (in the z-direction). The effect of viscous dissipation and axial conduction in the

fluid may be neglected.

(a) Write appropriate thermal boundary conditions at the top and bottom plates.

(b) Show that the bulk temperature of the fluid is given by

1

0

bT T dη= ∫ , where T is the temperature of the

fluid, y

Lη = , y is the coordinate normal to the plates, measured from the bottom plate.

(c) By integrating the energy equation across the channel, or otherwise, obtain an expression for the bulk

temperature gradient, bdT

dx.

(d) In the thermally developed region of the flow, w

b w

T T

T T

−−


η . Using this relation, show that the local heat transfer coefficient at the top plate, 0w b

qh

T T=

−, is a

constant in the thermally developed region. Here, ( )wT x is the temperature of the top plate.

(e) Using the result of part (d), show that (i) w bdT dT

dx dx= and (ii) b

T dT

x dx

∂=


region.

(f) Determine the temperature distribution in the fluid relative to the temperature of the top plate, in the

thermally developed region of the flow.

(g) Determine the Nusselt number at the top plate. [2+2+2+2+4+4+4=20 marks]


incompressible flow of a constant-property Newtonian Fourier fluid through a circular duct of radius R ,

with step change in wall temperature, given by , 0

( , ), 0

i

w

T zT R z

T z

2

iT and wT are constants. The inlet fluid temperature is iT . The axial velocity through the duct is given

by ( )22 1 *avw w r⎡ ⎤= −⎢ ⎥⎣ ⎦ , where * /r r R= , r is the radial coordinate and avw is the average velocity

through the duct. The non-dimensional temperature, w

w i

T T

T Tθ −=

−, downstream of the discontinuity in

wall temperature is given by 2

*

1

( ) nZ

n n

n

c f r eλθ

∞−

=

=∑ , where, *( )nf r is the solution of the equation, * 2 * * 2

* *( ) [1 ( ) ] 0n n n

dfdr r r f

dr drλ+ − = , with boundary condition (1) 0nf = , satisfying the regularity

condition (0) 0nf ′ = , /( )Z z RPe= , (2 ) /avPe w R α= , and α is the thermal diffusivity of the fluid. The important constants are given in the following table.

n nλ (1)n nc f ′−

1 2.7043644 1.49758

2 6.679032 1.08848

3 10.67338 0.92576

(a) Show that the non-dimensional bulk temperature is given by ( )1

2

0

4 1* * *b r r d rθ θ⎡ ⎤= −⎢ ⎥⎣ ⎦∫ .

(b) Show that ( )1

2* * * *

2

0

(1)1 ( ) nn

n

fr r f r dr

λ

′⎡ ⎤− = −⎢ ⎥⎣ ⎦∫ . (c) Using the result of parts (a) and (b), show that the non-dimensional bulk-temperature may be

expressed as 2

21

(1)4 n

Zn nb

n n

c fe

λθλ

∞−

=

′= − ∑ .

(d) Obtain an expression for the local Nusselt number. Express your answer in terms of the constants nc ,

nλ , Z and other appropriate quantities. (e) Determine the numerical value of the Nusselt number in the thermally developed region.

[4+3+5+5+3=20 marks]


constant-property Newtonian Fourier fluid over a semi-infinite flat plate aligned with the direction of a

uniform oncoming isothermal free-stream. The plate is kept at a constant temperature, wT , higher than the

ambient temperature T∞ . The boundary-layer energy equation,

22

2

1 ***

uu V Ec

Y Pr Yx Y

θ θ θ ⎛ ⎞∂ ∂ ∂ ∂+ = + ⎜ ⎟∂ ∂∂ ∂ ⎝ ⎠

, admits

a similarity solution of the form ( ;Pr, )G Ecθ η= , where ww

T T

T Tθ

∞

−=

−,

*2

Y

xη = , * /x x L= ,

1/ 2 *ReY y= , * /y y L= , * ( )u

u Fu

η∞

′= = , [ ]1/ 2*

1Re ( ) ( )

2

vV F F

u xη η η

∞

′= = − , Re /u L υ∞= ,

( )2

p w

uEc

c T T

∞

∞

=−

, ( )F η is the solution of the equation 0F F F′′′ ′′+ = , with boundary conditions

(0) (0) 0F F ′= = , ( ) 1F ′ ∞ = , T is the fluid temperature, u and v are the components of the fluid velocity in the x and y directions, x is the coordinate along the plate, y is the coordinate normal to the

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plate, u∞ is the free-stream speed, L is a reference length, υ , pc and Pr are respectively the kinematic viscosity, the specific heat capacity at constant pressure and the Prandtl number of the fluid. The origin of

the coordinate system is at the leading edge of the plate. The values of F , F ′and F ′′ at selected values of η are given in the table below.

η ( )F η ( )F η′ ( )F η′′ 0.0 0.0 0.0 0.46960

0.1 0.00235 0.04696 0.46956

0.2 0.00939 0.09391 0.46931

(a) Starting with the boundary-layer energy equation stated above, derive an ordinary differential equation

for G.

(b) Obtain appropriate boundary conditions for the equation derived in part (a).

(c) Show that the local Nusselt number predicted by the similarity solution is of the

form, (Pr, ) Remx xNu c Ec= , where Re /x u x υ∞= is the local Reynolds number, m is a universal constant and (Pr, )c Ec is a constant that depends on the Prandtl and Eckert numbers. Determine the

value of m.

(d) Obtain a suitable approximate form of the equation derived in part (a), which can be solved

analytically, for fluids with 1Pr . Do not attempt to solve this equation.

(e) For the special case when 1Pr and 1Ec , solve the approximate equation of part (d) without the

viscous dissipation term, and show that (Pr, ) Prnc Ec b= when 1Pr and 1Ec , where b and n are universal constants. Determine the values of b and n. [5+2+7+4+7=25 marks]

4. Consider steady laminar two-dimensional natural-convection boundary-layer flow of a Newtonian

Fourier fluid along a semi-infinite heated vertical flat plate kept at a constant temperature, wT , higher than

the ambient temperature, T∞ . The boundary-layer equations admit a similarity solution of the form

( )1/ 2* *0

4 ( )u

u x Fu

η′= = , ( )

[ ]1/ 4 1/ 4*

0

1( ) 3 ( )

4

vV Gr F F

u xη η η′= = − , ( )

w

T TG

T Tθ η∞

∞

−= =

−,

where * xx

L= , 0 ( )wu g T T Lβ∞ ∞= − ,

3 2( ) /wGr g T T Lβ υ∞ ∞= − , ( )F η and ( )G η are solutions of the

equations 23 2( ) 0F FF F G′′′ ′′ ′+ − + = , 3Pr ( ) 0G F Gη′′ ′+ = , subject to the boundary conditions

(0) (0) 0F F ′= = , (0) 1G = , ( ) 0F ′ ∞ = , ( ) 0G ∞ = , * 1/ 4(4 )

Y

xη = , 1/ 4 *Y Gr y= , * yy

L= , g is the

gravitational acceleration, β∞ is the coefficient of thermal expansion at the ambient temperature, υ is the kinematic viscosity of the fluid, u and v are the components of the fluid velocity in the x and y directions, T is the fluid temperature, x is the coordinate along the plate, pointing vertically upwards, y

is the coordinate normal to the plate and L is a reference length. The origin of the coordinate system is at

the leading edge of the plate. The following table gives the values of F, G and their derivatives for Pr = 1,

at selected values of η :

η ( )F η ( )F η′ ( )F η′′ ( )G η ( )G η′ 0 0 0 0.6421 1 -0.5671

0.1 0.0030 0.0593 0.5450 0.9433 -0.5669

3.6 0.5016 0.0254 -0.0321 0.0136 -0.0210

3.8 0.5061 0.0197 -0.0255 0.0009 -0.0155

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(a) Show that the local Nusselt number predicted by the similarity solution is of the form,

(Pr) mx xNu a Gr= , where xGr is the local Grashof number, m is a universal constant and (Pr)a is a constant that depends on the Prandtl number. Determine the value of m.

(b) Assume that the similarity solution gives a good prediction for natural convection over a large finite

plate of length L in the vertical direction and width b in the horizontal direction. Using the relation

between the local Nusselt number and the local Grashof number given in part (a), obtain an expression for

the total rate of heat transfer, Q, from the plate. Express your answer in terms of the constant, (Pr)a ,

defined in part (a), the Grashof number, Gr , and other relevant parameters.

(c) Using your answer to part (b), determine the value of the ratio L

L

Nu

Nu, where L

h LNu

k= is the average

Nusselt number, k is the thermal conductivity of the fluid and h is the average heat transfer coefficient

defined by ( )w

Qh

b L T T∞=

−.

(d) Show that the thickness of the heated layer varies with x as 1/ 4

n

T xGr cL L

δ − ⎛ ⎞= ⎜ ⎟⎝ ⎠

, where n is a

universal constant and c is a constant that depends on the Prandtl number. Determine the value of n.

(e) Use the table to determine the value of the constant (Pr)a when Pr = 1.

(f) The value of the constant, c, in part (d), depends on the criterion used to define the thickness of the

heated layer. Using the 99% criterion and the above table, estimate the value of c when Pr = 1.

(g) Consider the case when the plate is porous and a suction velocity ( )* *( ,0) dV x B x= − is applied at the wall, where B is a positive non-dimensional constant. Determine the value of d for which a similarity

solution of the form indicated above exists for natural-convection boundary-layer flow over a semi-

infinite vertical porous plate.

(h) Modify the boundary conditions on F and G to account for the effect of applied suction at the wall,

when d is equal to the value obtained in part (g). [6+6+2+3+2+2+2+2=25 marks]


(a) Consider steady laminar incompressible axisymmetric hydrodynamically and thermally developed

forced convection of a constant-property Newtonian Fourier fluid in a circular tube with uniform surface

heat flux for the case when viscous dissipation is negligible. The solution for the temperature field, of the

form ( ) ( ) ( )wT r,z T z f r= + , presented in most textbooks is obtained by assuming that axial conduction in the fluid is negligible. A student claims that neglecting the axial conduction term in the energy equation

does not introduce any error in the solution for the temperature field, even if the Peclet number of the

flow is small, as the heat flux in the axial direction is constant. Do you agree with the student ?

(b) Consider steady laminar forced-convection boundary-layer flow of a constant-property Newtonian Fourier fluid along a semi-infinite heated flat plate aligned with the direction of a uniform isothermal

oncoming free-stream. The plate is subjected to a constant surface heat flux. A student argues that if the

Prandtl number of the fluid is Pr = 1, the temperature field can be predicted from a knowledge of the

velocity field, without solving the energy equation, using Reynolds analogy. Do you agree with the

student ?

(c) Consider steady laminar natural convection heat transfer to air from a large heated isothermal rectangular vertical flat plate. An engineer thinks that the rate of heat transfer from the plate will be

doubled if the surface area of the plate is doubled. Two options of doubling the surface area of the plate

are available to him. In the first option, the length of the plate in the vertical direction is doubled, keeping

the width in the horizontal direction the same. In the second option, the width of the plate is the horizontal

direction is doubled, keeping the length in the vertical direction the same. Which of the two options

would you recommend to the engineer ? [4+3+3=10 marks]

END OF PAPER

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carefully. Make suitable assumptions wherever necessary. Please state your assumptions clearly. Clearly

indicate the coordinate system used in your analysis. The following information may be useful:


j j

jB R

aa n d A dV

x

∂=


1 (a) By integrating Cauchy’s equations of motion,j ii

i

j

Dvg

Dt x

τρ ρ

∂= +

∂, over a material region ( )mR t

bounded by a material surface ( )mS t , and using the relation,( ) ( )m mR t R t

d DdV dV

dt Dt

φρφ ρ=∫∫∫ ∫∫∫ , derive the following equation representing balance of mechanical energy for a closed system or material region of

fluid that is translating, rotating and deforming:

( ) ( )

,

1 2 3

m m

mechs

R t R t

dEW p dV dV

dt

Term Term Term

= + ∇⋅ − Φ∫∫∫ ∫∫∫v&

Here, ρ is the density of the fluid, iiDx

vDt

= is the component of the fluid velocity, v, in the ix direction,

ig is the component of the gravitational acceleration, g , in the direction of the coordinate axis ix , t is

the time,D

Dt denotes material derivative,

mech kin potE E E= + is the mechanical energy of the system,

( )

1

2m

kin i i

R t

E v v dVρ= ∫∫∫ is the kinetic energy of the system, ( )m

pot i i

R t

E g x dVρ= − ∫∫∫ is the gravitational

potential energy of the system,

( ) ( )( , , )

m m

s i i i ji j

S t S t

W S t v dA v n dAτ= =∫ ∫n r& is the net rate at which work is

done on the system by surface forces exerted by the surrounding fluid, 1 2 3x x x= + +1 2 3r e e e denotes the position vector of a point,

ie is the unit vector in the ix -direction, jn is the jth component of the unit

outward normal, n̂ , at a point on the material surface, p is the thermodynamic pressure, ji ijdπΦ = is

the viscous dissipation function, and i jτ , ijπ and 1

2

jiij

j i

vvd

x x

⎛ ⎞∂∂= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

are respectively the elements of

the stress tensor, the deviatoric stress tensor, and the deformation tensor or strain-rate tensor.

(b) By integrating the ‘internal energy form’ of the equation representing local balance of thermal

energy, De

pDt

ρ = −∇ ⋅ − ∇ ⋅ +Φq v , over a material region, show that the time rate of change of the

internal energy, int

( )mR t

E e dVρ= ∫∫∫ , of the system is

mech14.weebly.com

int

( ) ( ) ( )

ˆ ,

4 5 6

m m mS t R t R t

dEdA p dV dV

dt

Term Term Term

= − ⋅ − ∇⋅ + Φ∫∫ ∫∫∫ ∫∫∫q n v

where e is the specific internal energy and q is the heat flux vector.

(c) Identify the term or terms (Term 1 to Term 6) that contribute to the change in the total energy, that is,

the sum of the mechanical energy and the internal energy, of the system.

(d) Identify the term or terms (Term 1 to Term 6) that represent irreversible conversion of mechanical

energy to internal energy. Give reasons to justify your answer.

(e) Identify the term or terms (Term 1 to Term 6) that represent reversible conversion of mechanical

energy to internal energy or vice versa. Give reasons to justify your answer.

[11+3+2+2+2=20 marks]

2. Consider steady laminar forced convective heat transfer to a constant-property Newtonian Fourier

fluid in the gap between two large parallel horizontal plates that are separated by a distance L. The

bottom plate moves in its own plane, in the x-direction, with constant speed U . The top plate is

stationary. The velocity field is given by (1 / )u U y L= − , 0v = , 0w = , where u, v and w are the components of the fluid velocity in the x, y and z directions respectively, and y is the coordinate normal

to the walls, measured from the bottom plate. The bottom plate is insulated. The top plate is maintained

at a constant temperature 2T . Assume that the plates have infinite span in the z-direction. Considering the

effect of viscous dissipation, determine

(a) the temperature distribution, ( )T y , in the fluid,

(b) the Nusselt number,hL

Nuk

= , at the top plate, based on the temperature difference between the two

plates, where h is the heat transfer coefficient and k is the thermal conductivity of the fluid.

[15+5=20 marks]

3. Consider steady laminar axisymmetric fully developed flow of a constant-property Newtonian Fourier

fluid through a circular duct of inside radius R, surrounded by air at constant ambient temperature T∞ .

The axial velocity distribution is given by

2

22 (1 )av

rw w

R= − , where r is the radial coordinate and avw is

the average velocity. The temperature distribution is a function of the radial coordinate only, so that the

energy equation reduces to 1

( ) 0d dT

k rr dr dr

+Φ = , where 2

dw

drμ ⎛ ⎞Φ = ⎜ ⎟⎝ ⎠

is the viscous dissipation, μ is

the fluid viscosity and k is the thermal conductivity of the fluid. Rotational symmetry requires that

( )0 0dTdr

= at r = 0. Heat transfer from the pipe wall to the surrounding air may be represented by a

convective boundary condition, with constant effective external heat transfer coefficient eh . Determine

(a) the temperature distribution ( )T r , (b) the wall temperature wT , (c) the centre-line temperature 0T ,

(d) the Nusselt number, (2 )h R

Nuk

= , where 0

w

w

qh

T T=

−is the internal heat transfer coefficient, wq is

the heat flux into the fluid at the inside surface, r = R, of the duct, (e) the value of the Nusselt number for

the special limiting case when the Biot number,

(2 )eh RBik

= , is very large, that is, 1Bi >> .

[10+2+2+4+2=20 marks]

mech14.weebly.com





Correction:

Q3(e) should read “Determine the non-dimensional temperature 2

av

T T

w

k

θμ

∞−= for the special limiting case

when the Biot number,

(2 )eh RBik

= , is very large, that is, 1Bi >> .”





Correction:


av

T T

w

k

θμ



(2 )eh RBik

= , is very large, that is, 1Bi >> .”





Correction:


av

T T

w

k

θμ



(2 )eh RBik

= , is very large, that is, 1Bi >> . ”

mech14.weebly.com

1







state your assumptions clearly. The following information may be useful:

2

02

te dtπ∞ − =∫ ,

3

0

0.892979512te dt

∞− =∫ ,

21( ) ( ) ,zerfc z d z z erfc z e

π−= −∫ lim ( ) 0erfcη η η→∞ =


incompressible flow of a constant-property Newtonian Fourier fluid through a circular duct of radius a,

with step change in wall temperature, given by , 0

( , ), 0

i

w

T zT a z

T z

2

2. Consider steady laminar two-dimensional natural convection boundary-layer of ‘Hupnol’ along a large

vertical heated isothermal plate maintained at a temperature wT higher than the temperature T∞ of the

fluid at large distances from the plate. The fluid ‘Hupnol’ has the unusual property that perturbations of

its density about ρ∞ varies as the cube of perturbations of its temperature from T∞ , with smaller density corresponding to larger temperature, so that ( ) 3[1 ( ) ]T T Tρ ρ γ∞ ∞= − − , where γ is a positive constant. The effects of viscous dissipation are negligible.

(a) Write the boundary-layer momentum equations with a body force term ( )Tρ g, in the x and y directions, in dimensional form, where g is the gravitational acceleration, x is the vertical coordinate and y is

the horizontal coordinate measured from the plate. Consider the orientation of the x-axis to be such that the

positive x-direction is upwards, and the origin of the coordinate system is at the bottom edge of the plate.

(b) Write the boundary-layer energy equation in dimensional form.

(c) Write the continuity equation in dimensional form.

(d) Write appropriate boundary conditions for solving the differential equations of parts (a), (b) and (c).

(e) Using your answer to part (d), determine the pressure gradient inside the boundary-layer.

(f) Using your answer to part (e), obtain an expression for the buoyancy force per unit volume in the x-

direction, in terms of temperature, when the fluid is Hupnol.

(g) Use scale analysis to obtain estimates for the orders of magnitude, 0u ,

0δ , and 0v , respectively, of the

vertical component of the velocity at distance L from the leading edge of the plate, the thickness of the

boundary layer, and the horizontal component of the velocity, for natural convection of Hupnol.

(h) Non-dimensionalize the equations describing natural-convection boundary-layer flow of Hupnol using

the following non-dimensional variables: * xx

L= ,

0

y

Yδ

= ,

*

0

uu

u= ,

0

v

Vv

= , w

T T

T Tθ ∞

∞

−=

−.

(i) Write appropriate boundary conditions for the system of equations of part (h).

(j) The boundary-layer equations for natural convection of Hupnol along a semi-infinite flat plate admit a

similarity solution of the form ( )ax Fψ η= , = ( )bx Gθ η , satisfying the boundary conditions of part

(i), where ψ is the streamfunction defined by the equations *uY

ψ∂=∂

, *

Vx

ψ∂= −

∂, and

m

Y

xη = , for

some values of a, b and m. Determine these values of a, b, and m.

(k) The local heat transfer coefficient varies as nh Ax= , where A and n are constants. Using your answer

to part (j), determine the value of n.

[2+1+1+3+1+1+3+3+3+5+2= 25 marks]


low Prandtl number constant-property Newtonian Fourier fluid over a semi-infinite flat plate aligned with

the direction of a uniform isothermal oncoming flow with flow speed u∞ and temperature T∞ . The effect

of viscous dissipation is negligible. The plate is subjected to a constant surface heat flux, 0q . The

velocity field may be approximated by u u∞= , 0v = in the entire thermal boundary layer. Here, u and v are the components of the velocity in the x and y directions, x is a coordinate along the plate, measured

from the leading edge, y is the coordinate normal to the plate, measured from the plate.

(a) Write the boundary-layer energy equation using the approximate velocity field given above, in

dimensional form.

(b) By differentiating the equation of part (a), obtain a partial differential equation describing the spatial

variation of the heat flux,

y

Tq k

y

∂= −

∂, in the direction normal to the plate.

(c) Obtain suitable boundary conditions for the partial differential equation of part (b).

mech14.weebly.com

3

(d) Show that the solution of the partial differential equation of part (b), subject to the boundary

conditions obtained in part (c), is given by ( )0yq q erfc η= , where 2

y

x

u

ηα

∞

= , α is the thermal

diffusivity of the fluid, ( )erfc η is the complementary error function defined by ( ) 1 ( )erfc erfη η= − ,

and ( )erf η is the error function defined by 2

0

2( ) terf e dt

η

ηπ

−= ∫ . (e) Use your answer to part (d) to determine the temperature distribution ( , )T x y .

(f) Determine the variation of the wall temperature, ( )wT x , with distance along the plate.

(g) Obtain an expression for the local Nusselt number. Express your answer in terms of the local

Reynolds number and the Prandtl number. [1+2+3+9+4+2+4=25 marks]

4. A low Prandtl number constant-property Newtonian Fourier fluid enters the gap between two large

heated horizontal parallel plates. The distance between the plates is L. Both the plates are maintained at

constant temperature wT . The velocity distribution at the inlet is uniform. The speed of the fluid at the

inlet is U . The effects of viscous dissipation and axial conduction in the fluid may be neglected. For low Prandtl number fluids, the temperature profile in a duct develops more rapidly than the velocity profile. In

such a situation, the temperature profile can be determined based on a uniform velocity profile. Thus, the

temperature distribution may be obtained by solving the equation,

2

2

T TU

x yα∂ ∂=

∂ ∂, where α is the

thermal diffusivity of the fluid. This is called the “slug flow” solution. In the thermally developed region,

( )ww b

T TY

T Tφ− =

−, where bT is the bulk temperature of the fluid,

yY

L= , and y is the coordinate normal to

the plates, measured from the bottom plate.

(a) Show that in the thermally developed region, the energy equation reduces to

2

2( )Y

Y

ψ φ∂ = −∂

, where

w

b

T T

dTL Pe

dx

ψ −= , x is the coordinate along the plates and ULPeα

= .

(b) Write appropriate boundary conditions for solving the differential equation of part (a).

(c) Show that the non-dimensional bulk temperature, w bb

b

T T

dTL Pe

dx

ψ −= , may be determined using the

relation

1

0

b dYψ ψ= ∫ .

(d) Obtain a relation between φ , ψ , and bψ . (e) Consider the two-level successive approximation scheme where the non-dimensional temperature

( )kψ at the k-th iteration is determined by solving the equation,( )

( )2

1

2( )

kk

YY

ψ φ −∂ = −∂

, with appropriate

boundary conditions, and the non-dimensional temperature ( )kφ at the k-th iteration is updated using the

relation of part (d). Starting with the initial guess, ( ) ( )0 1Yφ = , carry out three steps of this iterative

mech14.weebly.com

4

scheme, that is, determine the functions ( ) ( )1 Yφ , ( ) ( )2 Yφ and ( ) ( )3 Yφ . Obtain estimates of the Nusselt

number at the lower wall at each of the three iterations, and state whether or not the iterations are

converging.

[2+2+2+2+12=20 marks]


(a) Consider steady laminar axisymmetric hydrodynamically and thermally developed forced convection

of water in a long circular tube. The inside surface of the tube is at a constant temperature higher than the

temperature of the fluid at the inlet. An engineer thinks that the heat flux at the surface of the tube in the

thermally developed region of the flow will be doubled if the flow rate is doubled. Do you agree with the

engineer ?

(b) Consider forced convection from a large heated isothermal plate aligned with the direction of a

uniform isothermal oncoming free-stream of air at low Mach numbers. An engineer wants to double the

rate of heat transfer from the plate by increasing the free-stream speed. By what factor should he increase

the free-stream speed in order to double the rate of heat transfer from the plate ?

(c) Consider steady laminar forced-convection boundary-layer flow of a constant-property Newtonian

Fourier fluid along a semi-infinite heated isothermal flat plate aligned with the direction of a uniform

isothermal oncoming free-stream, when the effects of viscous dissipation are important. A student argues

that if the Prandtl number of the fluid is Pr = 1, the wall heat flux can be predicted without solving the

energy equation, using the fact that with appropriate non-dimensionalization, the non-dimensional

temperature profile is identical to the non-dimensional velocity profile when Pr = 1. Do you agree with

the student ?

(d) Consider steady laminar incompressible axisymmetric hydrodynamically and thermally developed

forced convection of a constant-property Newtonian Fourier fluid in a circular tube with uniform surface

heat flux for the case when viscous dissipation is negligible. The solution for the temperature field, of the

form, ( ) ( ) ( )wT r,z T z f r= + , presented in most textbooks is obtained by assuming that axial conduction in the fluid is negligible. A student claims that neglecting the axial conduction term in the energy equation

does not introduce any error in the solution even if the Peclet number of the flow is small. Do you agree

with the student ?

(e) Convective heating or cooling may be described under certain circumstances by Newton’s law of

cooling, which states that the rate of heat loss of a body is directly proportional to the difference in

temperatures between the body and its surroundings. Does Newton’s law of cooling hold for natural

convection from a large vertical isothermal plate ?

[3+3+3+3+3=15 marks]

END OF PAPER

mech14.weebly.com




Sub. No.: ME60014 Sub. Name: Convective Heat and Mass Transfer

Instructions: Attempt all questions. Symbols have their usual meanings. Please explain your work carefully.

Clearly indicate the coordinate system used in your analysis. Make suitable assumptions wherever necessary.

Please state your assumptions clearly. The following information may be useful:

1) j

j j

jB R

aa n d A dV

x

∂=

∂∫ ∫ , where a is a continuous vector field in a region R bounded by a surface B.

2)

( ) ( )m mR t R t

d DdV dV

dt Dt

φρφ ρ=∫ ∫

1. Consider a material region of fluid, ( )mR t , subjected to a gravitational body force, that is translating, rotating and deforming continuously in a flow field with internal volumetric heat generation.

(a) Show that the rate at which work is done by surface forces exerted by the fluid surrounding the material

region,

( )msurf i ji j

B t

W v n dAτ= ∫& , may be expressed as 1 2 3 4surfW I I I I= + + +& , where ( )

1

mR t

DkI dV

Dtρ= ∫ ,

( )2

m

g

R t

DI dV

Dtρ

Ω= ∫ , 3

( )m

comp

R t

I w dV= ∫ & , 4( )mR t

I dV= Φ∫ , ρ is the density of the fluid,

1

2i ik v v= ,

g i ig xΩ = − , compw p= − ∇⋅v& , ,ji ijdπΦ = ig is the component of the gravitational acceleration, g , in the

direction of the coordinate axis ix , i

i

Dxv

Dt= is the component of the fluid velocity, v, in the ix -direction, t

is the time, j

j

Dv

Dt t x

∂ ∂≡ +∂ ∂

denotes material derivative, p is the thermodynamic pressure, i jτ , ijπ and

1

2

jiij

j i

vvd

x

Documents

mech14.weeblymech14.weebly.com/uploads/6/1/0/6/61069591/me_chmt... · stress vector at a point on a plane normal to n.Using the Gauss divergence theorem and your answer to part (a),