CHAPTER ONE

INTRODUCTION

1.1 General Background

Heat transfer deals with the study of the thermal energy transport phenomena within a medium

by molecular interaction, motion of the surrounding fluids, resulting from a spatial variation in

temperature. Heat transfer has a wide range of applications in many important areas, for instance,

thermal and nuclear power plants, heat shields for space vehicles, furnaces, electronic devices,

internal combustion engines, manufacturing industries etc. Generally, heat transfer takes place by

three different modes such as conduction, convection and radiation. Conduction is the

transportation of the thermal energy among the neighboring molecules in a substance due to the

temperature gradient, and the heat transfer takes place from a high temperature region to low

temperature region. Convective heat transfer occurs when heat is being transferred from one

place to another by the movement of the fluids, for example, heat transfer in liquids and gases.

Though Convection is referred as a separate method of heat transfer, it involves the combined

processes of conduction and advection. Radiation takes place when the heat is transferred due to

the emission of electromagnetic waves.

1.2 Heat Conduction

Heat conduction analysis requires the specification of temperatures, heat sources and heat flux in

the regions of the object. Usually, for classifying the temperature distributions three types of

coordinate system such as one-dimensional (1D), two-dimensional (2D) and three-dimensional

(3D) are considered. However, the transportation phenomena of the heat also depends upon the

nature of the conduction process. In steady state conduction process, the heat flow rate, heat

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fluxes, temperatures are considered to be independent of time. For the unsteady or transient state

analysis, the temperature is considered as a function of time. The solution of the current project

involves the 2D transient analysis which includes both the analytical and numerical solution.

1.3 Motivation of the Present Work

In the steel industries, heat transfer analysis is carried out to find the proper quenching time of

the steel ingots. The steel ingots came out of the blast furnace, and quenched quickly in a large

reservoirs of cold water. Therefore, it is critical to determine the quenching time so that the

temperature distribution remains uniform when they are taken out from the reservoir.

1.4 Objective

1. To find the temperature distribution inside a long rectangular ingot as it is quenched in a

cold liquid

2. To obtain the closed form analytical solutions by the separation of variable methods

3. To obtain the numerical solutions using finite difference method (explicit scheme)

4. To compare the analytical and numerical results of the temperature distributions

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CHAPTER 2

MATHEMATICAL MODELLING OF 2D TRANSIENT HEAT CONDUCTION PROBLEMS

2.1 Analytical Solution of 2D Transient Heat Transfer Phenomena of the Ingot

The ingot is subjected to convective cooling boundary conditions on each sides. The initial

temperature of the ingot is considered to be Ts, the heat transfer coefficient of the surrounding

fluid is h with a temperature of T∞ which is considered as constant. The other physical properties

thermal conductivity of the ingot (κ) and the thermal diffusivity (α) is also assumed to be

constant. Therefore, for the analysis we have the equations as follows:

Governing equation: ∂T/∂t = α* (∂2T/∂x2 + ∂2T/∂y2)

Boundary conditions: ± κ ∂T/∂x = h (Ts-T∞)

± κ ∂T/∂y = h (Ts-T∞)

Initial conditions: T (0, x, y) = To

2.2 Solution Procedure

This problem can be solved by using the direct separation of variable methods or the product of

two solutions. For this project the product of two solutions method have been used to find the

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h (Ts-T∞)

H/2

L/2

h (

Ts

-

h (Ts-T∞)

h (Ts-T∞)

h (Ts-T∞)

L

H

Figure 1 Analytical a nalysis of 2D transient heat transfer phenomena

∂T/∂x = 0

∂T/∂

y =

0

analytical solution. In this method, the generalized form of the analytical solution can be written

as Θ (x, y, t) = X(x, t) * Y(y, t)

For applying this method, origin of the axis was shifted to the center of the 2D geometry, and the

governing and boundary equations are re-written as follows:

Governing equation: ∂T/∂t = α* (∂2T/∂x2 + ∂2T/∂y2)

Boundary conditions: ∂T/∂x = 0 at x = 0

∂T/∂y = 0 at y = 0

- κ ∂T/∂x (L/2) = h (T(L/2)-T∞)

- κ ∂T/∂y = h (T(H/2)-T∞)

The final outcome of the solution can be written as:

X(t*,η) = ∑ 2sinλn/( λn+ sinλn cosλn)*exp[λn2*t*] * cosλn for n = 1 to ∞

Y(t*,η) = ∑ 2sinλn/( λn+ sinλn cosλn)*exp[λn2*t*] * cosλn for n = 1 to ∞

The non-dimensional definitions are:

η = x/(L/2) t* = (4*α/L2)*t for X-axis

η = y/(H/2) t* = (4*α/H2)*t for Y-axis

and λntanλn = Bi

Therefore, the final solution can be written as:

Θ (x,y,t) = X(t*,η) * Y(t*,η)

For the convenience of the solution, I have non-dimensionalized the temperature as follows:

Θ = T-T∞/ To -T∞

So the final temperature will be calculated from the following expression:

T (x,y) = Θ(x,y,t)*( To -T∞) + T∞

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I have written a Matlab code for solving the analytical solutions. For the exact solution we need

the Eigen values, and which were calculated by another simple program which was provided as a

part of the course contents.

2.3 Numerical Solution of 2D Transient Heat Transfer Phenomena of the Ingot

There are three main techniques available for solving the mathematical models of heat

conduction problems: finite differences (FD), finite elements (FE) and finite volume (FV)

methods. The fundamental principle of these methods is based upon the discretization of the heat

equation and then solving the algebraic problem. Discretization is executed by considering the

medium as constituted of a summation of cells or volumes of nodes of finite size. Usually, nodes

are linked with each cells, and produces a mesh of points, thereby. The separation between any

two nodes is defined as mesh spacing. The temperature at each cell is represented by the

temperature at the corresponding nodal location of the cell. For this project, the governing

equation is solved using the explicit scheme of the finite difference method. The explicit scheme

is conditionally stable which is counted as a limitation. The calculation of temperature at every

nodal point (T i, j) gives stable and meaningful results only when the following condition is met:

(α*∆t)*(1/∆x2 + 1/∆y2) ≤ ½

The value of ∆t is chosen arbitrarily so that the above conditions is met.

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2.4 Algebraic equations developed by the Explicit Scheme:

The algebraic equations that are developed for the heat transfer phenomena of the ingot are as

follows:

For the interior nodes:

T i, j n+1 = Fo* [T i+1

n, j + Ti-1n, j + T i, j+1

n + T i, j-1n] + [1-4*Fo]*Ti

n, j

For the boundary nodes:

Right Side: T i, j n+1 = Fo* [ 2*Ti-1, j + T i, j+1

+ T i, j-1 + 2*Bi*T∞] + [1-4*Fo – 2*Bi*Fo]*Ti, j

Left Side: T i, j n+1 = Fo* [ 2*T2, j + T i, j+1

+ T i, j-1 + 2*Bi*T∞] + [1-4*Fo – 2*Bi*Fo]*T1, j

Top Side: T i, j n+1 = Fo* [ 2*Ti, j-1 + T i+1, j

+ T i-1, j + 2*Bi*T∞] + [1-4*Fo – 2*Bi*Fo]*Ti, j

Bottom Side: T i, j n+1 = Fo* [ Ti+1, 1 + T i-1, 1

+ 2*T i, 2 + 2*Bi*T∞] + [1-4*Fo – 2*Bi*Fo]*Ti, 1

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T i, j

Corner Nodes

Boundary Nodes

Inner Nodes

Figure 2 Numerical Grid for the 2D ingot

Corner Nodes:

Bottom Left: T 1, 1 n+1 = Fo* [ 2*T2, 1 + T 1, 2

+ 4*Bi*T∞] + [1-4*Fo – 4*Bi*Fo]*T1, 1

Bottom Right: T i, 1 n+1 = Fo* [ 2*Ti-1, 1 + 2*T i, 2

+ 4*Bi*T∞] + [1-4*Fo – 4*Bi*Fo]*Ti, 1

Top Right: T i, j n+1 = Fo* [ 2*Ti-1, j + 2*T i, j-1

+ 4*Bi*T∞] + [1-4*Fo – 4*Bi*Fo]*Ti, j

Top Left: T 1, j n+1 = Fo* [ 2*T2, j + 2*T 1, j-1

+ 4*Bi*T∞] + [1-4*Fo – 4*Bi*Fo]*T1, j

2.5 Parameters for the Calculation:

The physical properties of AISI 4000 Series Steel [1] are considered for the calculation:

Density = 7850 Kg/m3; Thermal conductivity = 52 W/m-K; Specific Heat = 4770 KJ/Kg-K

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0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

734.0

739.4

744.8

750.2

755.6

761.0

766.4

771.8

777.2

Temperature ()@ t = 400 sec

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

586.9

590.5

594.0

597.6

601.2

604.8

608.4

611.9

615.5

Temperature (K)@ t = 800 sec

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

489.7

492.0

494.4

496.7

499.1

501.5

503.8

506.2

508.6

Temperature (K)@ t = 1200 sec

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

425.4

426.9

428.5

430.0

431.6

433.2

434.7

436.3

437.9

Temperature (K)@ t = 1600 sec

CHAPTER 3

RESULTS AND DISCUSSIONS

3.1 Results: The temperature distribution from the numerical solution plotted below at five

different time scales:

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0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

382.9

383.9

384.9

386.0

387.0

388.0

389.1

390.1

391.1

Temperature() @ t = 2000 sec

Figure 3 Temperature Distribution of the ingot quenched in solution at different time steps for t = 400, 800, 1200, 1600 & 2000 secs respectively

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

0.8430

0.8647

0.8864

0.9081

0.9297

0.9514

0.9731

0.9948

1.017

Error Distribution (%)@t = 400 sec

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

1.467

1.479

1.492

1.505

1.518

1.531

1.544

1.557

1.569

Error Distribution(%)@ t = 800 sec

The error distribution contour plot between the analytical and numerical solution are as follows:

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0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

1.790

1.795

1.800

1.804

1.809

1.814

1.819

1.823

1.828

Error Distribution(%)@ t = 1200 sec

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)W

idth

(m)

1.845

1.847

1.848

1.850

1.851

1.853

1.854

1.856

1.857

Error Distribution (%)@ t = 1600 sec

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

1.696

1.702

1.707

1.712

1.718

1.723

1.729

1.734

1.740

Error Distribution (%)@ t = 2000 sec

Figure 4 Error distribution (%) between analytical and numerical solution over the geometry at different time steps for t = 400, 800, 1200, 1600 & 2000 secs respectively

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0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

311.1

311.4

311.7

312.0

312.3

312.6

312.9

313.1

313.4

For h = 200 W/m2/KT

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

250.6

251.9

253.1

254.4

255.6

256.9

258.1

259.4

260.7

For h = 100 W/m2/KT = 150 K

Figure 5 Temperature distribution in ingot for h = 200 W/m2/°K and T∞=300 °K

Figure 6 Temperature distribution in ingot for h = 100 W/m2/°K and T∞=150 °K

3.2 Discussions:

The value of external convective heat transfer coefficient and the initial temperature solution was

changed to see the effects on the temperature distribution at t = 2000 seconds

Case A: For h = 200 W/m2/°K and T∞ = 300 °K

Case B: For h = 100 W/m2/°K and T∞ = 150 °K

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0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

Length (m)

Wid

th (m

)

163.5

163.9

164.2

164.6

164.9

165.3

165.6

166.0

166.3

For h = 200 W/m2/KT = 150

Figure 7 Temperature distribution in ingot for h = 200 W/m2/°K and T∞=150 °K

Case C: For h = 200 W/m2/°K and T∞ = 150 °K

From the case A we can observe that if we increase the convective heat transfer co-efficient by

twice, the temperature reaches to ~ 313.4 °K to 311.1 °K after 2000 seconds. The temperature

was around 391.1 °K to 382.9 °K for the given conditions after the same amount time. So the

convective heat transfer coefficient plays an important role for the heat transfer phenomena. On

the other hand, if we fix the h at the given conditions and decrease the solution temperature by

half we can see that the temperature of the ingot reaches at ~260.7 °K to 250.6 °K after the same

amount of time. The temperature difference between the ingot and the surroundings is around

100 °K, and from that we can say that it is more important to use such a fluid which has a higher

heat transfer coefficient. From the case C we do not see any significant variation in the heat

transfer phenomena from the case A. Therefore, it is recommended to use a fluid of higher heat

transfer coefficient.

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From the error distribution pot we can see that the analytical and numerical solution had a good

agreement with an error of ~1.5%.

Due to lack of understanding of the implicit scheme and coding experience, I could not do the

numerical solution by implicit method as it is unconditionally stable. I suppose the implicit

scheme is much faster technique to solve the problem numerically. For the numerical solution I

choose the value of time steps (dt) arbitrarily which is not a good programming practice I guess.

There should much smarter way to select the value of time steps so that it can met the stability

condition. The program takes huge amount of time when I increase the number of nodes and

reduce the time steps.

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CHAPTER FOUR

CONCLUSIONS AND RECOMMENDATIONS

4.1 Conclusions:

The performance of cooling depends effectively on the convective heat transfer coefficient of

the solution. Though the lower ambient temperature contributes for cooling down the ingot,

but it is important to select a fluid of higher convective heat transfer coefficient. Also the

economic factors should be considered for choosing the solution.

4.2 Recommendations

The numerical solution should be done with the implicit scheme so that we don’t have to be

cautious about the value of the time steps. There is a scope of designing the numerical code

in a more efficient way so that it takes less computing time.

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REFERENCES

[1] Overview of materials for AISI 4000 Series Steel. MatWeb Material Property Data.

Retrieved from http://www.matweb.com

[2] Incropera, Frank P. DeWitt, David P. Fundamentals of Heat and Mass Transfer Fifth Edition,

R. R. Donnelley & Sons Company. 2002 John Wiley & Sons, Inc

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