##### Document Text Contents

Page 1

/f^

UCRL-52863

Conduction heat transfer solutions

James H. VanSant

u

March 1980

.:'•• •:•.•! i s UHUMITEE

Page 2

CONTENTS

Preface v

Nomenclature vii

Introduction . 1

Steady-State Solutions

1. Plane Surface - Steady State

1.1 Solids Bounded by Plane Surfaces 1-1

1.2 Solids Bounded by Plane Surfaces

—With Internal Heating 1-27

2. Cylindrical Surface - Steady State

2.1 Solids Bounded by Cylindrical Surfaces

—No Internal Heating 2-1

2.2 Solids Bounded by Cylindrical Surfaces

—With Internal Heating 2-33

3. Spherical Surface - Steady State

3.1 Solids Bounded by Spherical Surfaces

—No Internal Heating 3-1

3.2 Solids Bounded by Spherical Surfaces

—With Internal Heating 3-10

4. Traveling Heat Sources

4.1 Traveling Heat Sources 4-1

5. Extended Surface - Steady State

5.1 Extended Surfaces—No Internal Heating . • 5-1

5.2 Extended Surfaces—With Internal Heating 5-31

Transient Solutions

6. Infinite Solids - Transient

6.1 Infinite Solids—No Internal Heating 6-1

6.2 Infinite Solids—With Internal Heating 6-22

7. Semi-Infinite Solids - Transient

7.1 Semi-Infinite Solids—No Internal Heating 7-1

7.2 Semi-Infinite Solids—With Internal Heating . . . . 7-22

8. Plane Surface - Transient

8.1 Solids Bounded by Plane Surfaces

—No Internal Heating 8-1

8.2 Solids Bounded by Plane Surfaces

—With Internal Heating 8-52

iii

Page 281

Section 8.2. Solids Bounded by Plane Surfaces—With Internal Heating.

Case Mo. References Description Solution

8 .2 .8 9, p. 311 In f in i t e p late with

time-dependent heating,

t = t , 0 < x < H, T = 0.

t = t^, x = 0, A, T > 0.

q . . . = fr8/2,

s - - 1 , 0, 1, 2 , . . . .

00

1

to

\*-l^\

(t - t Q )k

Bar l+s /2 " (1 + s/2)

! l - r/(2 + s /2)2 s+2

n=0 L

8.2 .9 9, p. 404 In f in i t e p late with vari

able internal heating,

t = t Q , 0 < x < SL, T = 0.

t = t , x = S., T > 0.

q'" = q^" + 3 ( t - t „ ) .

q = 0, X = 0, X >_ 0.

• x

:

<*- y k if.

«^2 " B l

c o s ( V l x ) 1

cos (VI )

16 V (-1)" exp {[-(2n + 1) V + 4 B ] Fo/4f cos L(2n + DTTX/2.

+ * ^ f4B - (2n + 1) V ] (2n + 1)

n=0 L

B = eavk

Page 282

Section 8.2. Solids Bounded by Plane Surfaces—With Internal Heating.

Case No. References Description Solution

8.2.10 9, p. 405 Case 8.2.9 with

convection boundary

h, t Q at x = I.

(t- tQ)k Bi cos (Vlx)

l l l n 2 ~ BTBi cos (V"B) -iTS sin (V~B) ~ B «0 * L

+ 2 Bi

™ cos (XnX) exp [(B - Xfl) Fo ]

„ (B - \ 2 ) \\2 + Bi(Bi + 1) cos (XJ n=0 V n/ L n J n

X tan (X ) = Bi, B = 3A A n n

8.2.11 19,

p. 3-29

Infinite plate of infinite G Jin [l - (9/Gf) + 6 = LH/C

conductivity, variable

specific heat, convection

boundaries and steady heating.

0. t = t , 0 < x < 21, T

c = c Q + B ( t - t Q ) .

n

o

%

k-t-oo

-h,tf

G = (1 + q £ " X.)/hL(t - t f ) , C = .1 - c f)/c Q, H = hc/pcA

H evaluated at t Q , c f evaluated at t f

I. = R. x (surface area/volume) = 1

9 = (t - t Q)/(t f - t Q)

\^2i.*\

Page 561

65. D. R. Burnett, "Transient Temperature Measurement Errors in Heated Slabs

for Thermocouples Located at the Insulated Surfaces," J. Heat Transfer 83,

505-506 (1961).

66. J. J. Brogan and P. J. Scheider, "Heat Conduction in a Series Composite

Wall," J. Heat Transfer 83, 506-507 (1961).

67. M. R. El-Saden, "Heat Conduction in an Eccentrically Hollow Infinitely

Long Cylinder with Internal Heat Generation," J. Heat Transfer 83, 510-512

(1961).

68. R. S. Fairall, R. A. Belcher, "Unsteady-State Heat Transfer in Solids with

Radiation at One Boundary," J. Heat Transfer 84, 266-267 (1962).

69. Y. T. Tsui and F. K. Tsou, "Ratio of Radial to Total Heat Flow in a

Circular Rod," J. Heat Transfer 85, 285-286 (1963).

70. G. Horvay and M. Dacosta, "Temperature Distribution in a Cylindrical Rod

Moving From a Chamber at One Temperature to a Chamber at Another

Temperature," J. Heat Transfer 86, 265-270 (1964).

71. F. Jackson, "Moving Heat Sources with Change of Phase," J. Heat Transfer

87., 329-339 (1965).

72. A. Hiji-Sheikh and E. M. Sparrow, "Transient Heat Conducti.on in a Prolate

Spheroidal Solid," J. Heat Transfer 88, 331-333 (1966).

73. J. C. Rowley and J. B. Payne, "Steady State Temperature Solution for a

Heat Generating Circular Cylinder Cooled by a Ring of Holes," J. Heat

Transfer Stf, 531-536 (1964).

74. A. V. Luikov, Analytical Heat Diffusion Theory (Academic Press, Mew York,

1968).

75. R. B. Stewart and V. J. Johnson, A Compendium of the Properties of

Materials at Low Temperatures, National Bureau of Standards, Boulder,

WADD Tech Report 60-56, Part IV (1961).

76. Handbook on Materials for Superconducting Machinery (Metals and Ceramics

Information Center, Battelle, Columbus, Jan. 1977).

77. K. J. Mody and M. N. Ozisik, "Melting of a Semi-Infinite Solid with

Non-Linear Heat Transfer at the Boundary," Letters in Heat and Mass

Transfer, 2, 487-494 (1975).

78. B. T. F. Chung and L. T. Yeh, "Freezing and Melting of Materials with

Variable Properties and Arbitrary Heat Fluxes," AIAA Journ., 14, No. 3,

388-390 (1976).

R-5

Page 562

79. E. Hahne and V. Grigull, "Formfaktor Und Form-widerstand Der Stationaren

Mehrdimensionalen Warmeleltung," Int. J. Heat Mass Transfer, 18,

751-767 (1975).

80. A. A. Irlev, "Steady State Heat Conduction in an Eccentric Annulus,"

Heat Transfer-Soviet Research, J3, no. 3, 164-171 (1976).

81. A. Muzzio, "Approximate Solution for Corrective Fins with Variable

Thermal Conductivity," J. Heat Transfer, 680-682 (1976).

82. J. J. Brogan and P. J. Schneider, "Heat Conduction in a Series Composite

Wall," J. Heat Transfer, 506-508, Nov. 1961.

83. C. Kippenham and D. F. Schnell, "A Note on Heat Transfer Through Sheet

Fins," J. Heat Transfer, 293-295 (1964).

84. G. C. Sin, "Heat Conduction in the Infinite Medium with Lines of

Discontinuities," J. Heat Transfer, 293-298 (1965).

85. W. Fagan and S. Leipziger, "Nonuniform Cooling of a Heat-Generating

Cylinder or Sphere," J. Heat Transfer, 257-264, Aug. 1966.

86. G. E. Schneider, "Thermal Resistance of a Cylinder with Two Diametrically

Opposite, Symmetrical Isothermal Caps," J. Heat Transfer, 465-467, 1975.

87. E. I. Griggs, D. R. Pitts, and A. B. Goyal, "Conductive Shape Factors for

a Circular Cylinder Centered in a Rectangular Slab Having One and Two

Adiabatic Surfaces," J. Heat Transfer, 129-130 (1973).

88. R. A. Schmitz, "Heat Flux Through a Strip Heated Flat Plate," J. Heat

Transfer, 201-202 (1970).

89. A. M. Poindexter, "Temperature Charts for Simple Shapes in a Linearly

Changing Environment," J. Heat Transfer, 562-564 (1970).

90. F. M. Young and C. R. Savino, "Time-Temperature Charts for One-Dimensional

Conduction with Uniform Internal Heat Generation," J. Heat Transfer,

207-210 (1970).

91. P. J. Smith and J. Sucec, "Efficiency of Circular Fins of Triangular

Profile," J. Heat Transfer, 181-182 (1969).

92. J. H. Van Sant, Lawrence Livermore National Laboratory, unpublished work.

R - 6 . U.S. Government Printing Office: 1980/10-798-002/5513

/f^

UCRL-52863

Conduction heat transfer solutions

James H. VanSant

u

March 1980

.:'•• •:•.•! i s UHUMITEE

Page 2

CONTENTS

Preface v

Nomenclature vii

Introduction . 1

Steady-State Solutions

1. Plane Surface - Steady State

1.1 Solids Bounded by Plane Surfaces 1-1

1.2 Solids Bounded by Plane Surfaces

—With Internal Heating 1-27

2. Cylindrical Surface - Steady State

2.1 Solids Bounded by Cylindrical Surfaces

—No Internal Heating 2-1

2.2 Solids Bounded by Cylindrical Surfaces

—With Internal Heating 2-33

3. Spherical Surface - Steady State

3.1 Solids Bounded by Spherical Surfaces

—No Internal Heating 3-1

3.2 Solids Bounded by Spherical Surfaces

—With Internal Heating 3-10

4. Traveling Heat Sources

4.1 Traveling Heat Sources 4-1

5. Extended Surface - Steady State

5.1 Extended Surfaces—No Internal Heating . • 5-1

5.2 Extended Surfaces—With Internal Heating 5-31

Transient Solutions

6. Infinite Solids - Transient

6.1 Infinite Solids—No Internal Heating 6-1

6.2 Infinite Solids—With Internal Heating 6-22

7. Semi-Infinite Solids - Transient

7.1 Semi-Infinite Solids—No Internal Heating 7-1

7.2 Semi-Infinite Solids—With Internal Heating . . . . 7-22

8. Plane Surface - Transient

8.1 Solids Bounded by Plane Surfaces

—No Internal Heating 8-1

8.2 Solids Bounded by Plane Surfaces

—With Internal Heating 8-52

iii

Page 281

Section 8.2. Solids Bounded by Plane Surfaces—With Internal Heating.

Case Mo. References Description Solution

8 .2 .8 9, p. 311 In f in i t e p late with

time-dependent heating,

t = t , 0 < x < H, T = 0.

t = t^, x = 0, A, T > 0.

q . . . = fr8/2,

s - - 1 , 0, 1, 2 , . . . .

00

1

to

\*-l^\

(t - t Q )k

Bar l+s /2 " (1 + s/2)

! l - r/(2 + s /2)2 s+2

n=0 L

8.2 .9 9, p. 404 In f in i t e p late with vari

able internal heating,

t = t Q , 0 < x < SL, T = 0.

t = t , x = S., T > 0.

q'" = q^" + 3 ( t - t „ ) .

q = 0, X = 0, X >_ 0.

• x

:

<*- y k if.

«^2 " B l

c o s ( V l x ) 1

cos (VI )

16 V (-1)" exp {[-(2n + 1) V + 4 B ] Fo/4f cos L(2n + DTTX/2.

+ * ^ f4B - (2n + 1) V ] (2n + 1)

n=0 L

B = eavk

Page 282

Section 8.2. Solids Bounded by Plane Surfaces—With Internal Heating.

Case No. References Description Solution

8.2.10 9, p. 405 Case 8.2.9 with

convection boundary

h, t Q at x = I.

(t- tQ)k Bi cos (Vlx)

l l l n 2 ~ BTBi cos (V"B) -iTS sin (V~B) ~ B «0 * L

+ 2 Bi

™ cos (XnX) exp [(B - Xfl) Fo ]

„ (B - \ 2 ) \\2 + Bi(Bi + 1) cos (XJ n=0 V n/ L n J n

X tan (X ) = Bi, B = 3A A n n

8.2.11 19,

p. 3-29

Infinite plate of infinite G Jin [l - (9/Gf) + 6 = LH/C

conductivity, variable

specific heat, convection

boundaries and steady heating.

0. t = t , 0 < x < 21, T

c = c Q + B ( t - t Q ) .

n

o

%

k-t-oo

-h,tf

G = (1 + q £ " X.)/hL(t - t f ) , C = .1 - c f)/c Q, H = hc/pcA

H evaluated at t Q , c f evaluated at t f

I. = R. x (surface area/volume) = 1

9 = (t - t Q)/(t f - t Q)

\^2i.*\

Page 561

65. D. R. Burnett, "Transient Temperature Measurement Errors in Heated Slabs

for Thermocouples Located at the Insulated Surfaces," J. Heat Transfer 83,

505-506 (1961).

66. J. J. Brogan and P. J. Scheider, "Heat Conduction in a Series Composite

Wall," J. Heat Transfer 83, 506-507 (1961).

67. M. R. El-Saden, "Heat Conduction in an Eccentrically Hollow Infinitely

Long Cylinder with Internal Heat Generation," J. Heat Transfer 83, 510-512

(1961).

68. R. S. Fairall, R. A. Belcher, "Unsteady-State Heat Transfer in Solids with

Radiation at One Boundary," J. Heat Transfer 84, 266-267 (1962).

69. Y. T. Tsui and F. K. Tsou, "Ratio of Radial to Total Heat Flow in a

Circular Rod," J. Heat Transfer 85, 285-286 (1963).

70. G. Horvay and M. Dacosta, "Temperature Distribution in a Cylindrical Rod

Moving From a Chamber at One Temperature to a Chamber at Another

Temperature," J. Heat Transfer 86, 265-270 (1964).

71. F. Jackson, "Moving Heat Sources with Change of Phase," J. Heat Transfer

87., 329-339 (1965).

72. A. Hiji-Sheikh and E. M. Sparrow, "Transient Heat Conducti.on in a Prolate

Spheroidal Solid," J. Heat Transfer 88, 331-333 (1966).

73. J. C. Rowley and J. B. Payne, "Steady State Temperature Solution for a

Heat Generating Circular Cylinder Cooled by a Ring of Holes," J. Heat

Transfer Stf, 531-536 (1964).

74. A. V. Luikov, Analytical Heat Diffusion Theory (Academic Press, Mew York,

1968).

75. R. B. Stewart and V. J. Johnson, A Compendium of the Properties of

Materials at Low Temperatures, National Bureau of Standards, Boulder,

WADD Tech Report 60-56, Part IV (1961).

76. Handbook on Materials for Superconducting Machinery (Metals and Ceramics

Information Center, Battelle, Columbus, Jan. 1977).

77. K. J. Mody and M. N. Ozisik, "Melting of a Semi-Infinite Solid with

Non-Linear Heat Transfer at the Boundary," Letters in Heat and Mass

Transfer, 2, 487-494 (1975).

78. B. T. F. Chung and L. T. Yeh, "Freezing and Melting of Materials with

Variable Properties and Arbitrary Heat Fluxes," AIAA Journ., 14, No. 3,

388-390 (1976).

R-5

Page 562

79. E. Hahne and V. Grigull, "Formfaktor Und Form-widerstand Der Stationaren

Mehrdimensionalen Warmeleltung," Int. J. Heat Mass Transfer, 18,

751-767 (1975).

80. A. A. Irlev, "Steady State Heat Conduction in an Eccentric Annulus,"

Heat Transfer-Soviet Research, J3, no. 3, 164-171 (1976).

81. A. Muzzio, "Approximate Solution for Corrective Fins with Variable

Thermal Conductivity," J. Heat Transfer, 680-682 (1976).

82. J. J. Brogan and P. J. Schneider, "Heat Conduction in a Series Composite

Wall," J. Heat Transfer, 506-508, Nov. 1961.

83. C. Kippenham and D. F. Schnell, "A Note on Heat Transfer Through Sheet

Fins," J. Heat Transfer, 293-295 (1964).

84. G. C. Sin, "Heat Conduction in the Infinite Medium with Lines of

Discontinuities," J. Heat Transfer, 293-298 (1965).

85. W. Fagan and S. Leipziger, "Nonuniform Cooling of a Heat-Generating

Cylinder or Sphere," J. Heat Transfer, 257-264, Aug. 1966.

86. G. E. Schneider, "Thermal Resistance of a Cylinder with Two Diametrically

Opposite, Symmetrical Isothermal Caps," J. Heat Transfer, 465-467, 1975.

87. E. I. Griggs, D. R. Pitts, and A. B. Goyal, "Conductive Shape Factors for

a Circular Cylinder Centered in a Rectangular Slab Having One and Two

Adiabatic Surfaces," J. Heat Transfer, 129-130 (1973).

88. R. A. Schmitz, "Heat Flux Through a Strip Heated Flat Plate," J. Heat

Transfer, 201-202 (1970).

89. A. M. Poindexter, "Temperature Charts for Simple Shapes in a Linearly

Changing Environment," J. Heat Transfer, 562-564 (1970).

90. F. M. Young and C. R. Savino, "Time-Temperature Charts for One-Dimensional

Conduction with Uniform Internal Heat Generation," J. Heat Transfer,

207-210 (1970).

91. P. J. Smith and J. Sucec, "Efficiency of Circular Fins of Triangular

Profile," J. Heat Transfer, 181-182 (1969).

92. J. H. Van Sant, Lawrence Livermore National Laboratory, unpublished work.

R - 6 . U.S. Government Printing Office: 1980/10-798-002/5513