Transient heat conduction

HEAT AND MASS TRANSFER

REPORT ON

TRANSIENT HEAT CONDUCTION

Submitted by
CDT ARSHAD ALI
CDT NIKHIL BHATIA

Submitted to
Associate Proffessor Pradeep Kumar Singh

Indian Naval Academy

Knowledge Begets immortality

Certificate

This is to certify that this project report entitiled “TRANSIENT HEAT CONDUCTION” by CDT ARSHAD ALI and CDT NIKHIL BHATIA during the academic year 2010-2014, is a bonafide record of work carried out under my guidance and supervision.

- Associate Proffessor Pradeep Kumar Singh

Acknowledgement

We would like to express our sincere gratitude to our project guide “Associate Proffessor Pradeep Kumar Singh” for giving us the opportuity to work on this topic .It would never have been possible for us to take the project to the level without his innovative ideas and his continuous support and encouragement.

CDT ARSHAD ALI
CDT NIKHIL BHATIA

Abstract

Table of Contents

Chapter 1 -Introduction

Chapter 2 -Lumped System Analysis

Chapter 3 -Transient Heat Conduction in Large Plane Walls, Long Cylinders, and Spheres with Spatial Effects

Chapter 4 -Transient Heat Conduction in Semi-Infinite Solids

Chapter 5 -Transient Heat Conduction in Multidimensional Systems

Chapter 6 -Topic of Special Interest: Refrigeration and Freezing of Foods

Chapter 7 -Conclusion

Chapter 1
Introduction:

The temperature of a body, in general, varies with time as well as position. In rectangular coordinates, this variation is expressed as T(x, y, z, t), where (x, y, z) indicate variation in the x-, y-, and z-directions, and t indicates variation with time. In the preceding chapter, we considered heat conduction under steady conditions, for which the temperature of a body at any point does not change with time. This certainly simplified the analysis, especially when the temperature varied in one direction only, and we were able to obtain analytical solutions. In this chapter, we consider the variation of tem-perature with time as well as position in one- and multidimensional systems. We start this chapter with the analysis of lumped systems in which the temperature of a body varies with time but remains uniform throughout at any time. Then we consider the variation of temperature with time as well as position for one-dimensional heat conduction problems such as those associated with a large plane wall, a long cylinder, a sphere, and a semi-infinite medium using transient temperature charts and analytical solutions. Finally, we consider transient heat conduction in multidimensional systems by utilizing the product solution.

Chapter 2

Lumped System Analysis:

In heat transfer analysis, some bodies are observed to behave like a “lump” whose interior temperature remains essentially uniform at all times during a heat transfer process. The temperature of such bodies can be taken to be a function of time only, T(t). Heat transfer analysis that utilizes this idealization is known as lumped system analysis, which provides great simplification in certain classes of heat transfer problems without much sacrifice from accuracy.
Consider a small hot copper ball coming out of an oven (Fig. 4–1). Measurements indicate that the temperature of the copper ball changes with time, but it does not change much with position at any given time. Thus the temperature of the ball remains nearly uniform at all times, and we can talk about the temperature of the ball with no reference to a specific location. Now let us go to the other extreme and consider a large roast in an oven. If you have done any roasting, you must have noticed that the temperature distribution within the roast is not even close to being uniform. You can easily verify this by taking the roast out before it is completely done and cutting it in half. You will see that the outer parts of the roast are well done while the center part is barely warm. Thus, lumped system analysis is not applicable in this case. Before presenting a criterion about applicability of lumped system analysis, we develop the formulation associated with it. Consider a body of arbitrary shape of mass m, volume V, surface area As, density r, and specific heat cp initially at a uniform temperature Ti (Fig. 4–2). At time t ϭ 0, the body is placed into a medium at temperature Tρ, and heat transfer takes place between the body and its environment, with a heat transfer coefficient h. For the sake of discussion, we assume that Tρ Ͼ Ti, but the analysis is equally valid for the opposite case. We assume lumped system analysis to be applicable, so that the temperature remains uniform within the body at all times and changes with time only, T ϭ T(t). During a differential time interval dt, the temperature of the body rises by a differential amount dT. An energy balance of the solid for the time interval dt can be expressed as

Heat transfer into the body during dt = The increase in the energy of the body during dt hAs(Tρ - T) dt ϭ mcp dT
Noting that m ϭ rV and dT ϭ d(T Ϫ Tρ) since Tρ ϭ constant, Eq. 4–1 can be rearranged as

\

is a positive quantity whose dimension is (time)-1. The reciprocal of b has time unit (usually s), and is called the time constant. Equation 4–4 is plotted in Fig 4–3 for different values of b. There are two observations that can be made from this figure and the relation above:
1. Equation 4–4 enables us to determine the temperature T(t) of a body at time t, or alternatively, the time t required for the temperature to reach a specified value T(t).
2. The temperature of a body approaches the ambient temperature Tρ exponentially. The temperature of the body changes rapidly at the beginning, but rather slowly later on. A large value of b indicates that the body approaches the environment temperature in a short time. The larger the value of the exponent b, the higher the rate of decay in temperature.
Note that b is proportional to the surface area, but inversely proportional to the mass and the specific heat of the body. This is not surprising since it takes longer to heat or cool a larger mass, especially when it has a large specific heat.

Once the temperature T(t) at time t is available from Eq. 4–4, the rate of con- vection heat transfer between the body and its environment at that time can be determined from Newton’s law of cooling as
·
Q (t) = hAs[T(t) - Tρ]

The total amount of heat transfer between the body and the surrounding medium over the time interval t ϭ 0 to t is simply the change in the energy content of the body:
Q = mcp[T(t) - Ti]

The amount of heat transfer reaches its upper limit when the body reaches the surrounding temperature Tρ. Therefore, the maximum heat transfer between the body and its surroundings is (Fig. 4–4)
Qmax = mcp(Tρ - Ti)

We could also obtain this equation by substituting the T(t) relation from Eq. 4–4 into the Q (t) relation in Eq. 4–6 and integrating it from t ϭ 0 to t → ρ.

Criteria for Lumped System Analysis :

The lumped system analysis certainly provides great convenience in heat transfer analysis, and naturally we would like to know when it is appropriate to use it. The first step in establishing a criterion for the applicability of the lumped system analysis is to define a characteristic length as

When a solid body is being heated by the hotter fluid surrounding it (such as a potato being baked in an oven), heat is first convected to the body and subsequently conducted within the body. The Biot number is the ratio of the internal resistance of a body to heat conduction to its external resistance to heat convection. Therefore, a small Biot number represents small resistance to heat conduction, and thus small temperature gradients within the body. Lumped system analysis assumes a uniform temperature distribution throughout the body, which is the case only when the thermal resistance of the body to heat conduction (the conduction resistance) is zero. Thus, lumped system analysis is exact when Bi ϭ 0 and approximate when Bi >= 0. Of course, the smaller the Bi number, the more accurate the lumped system analysis. Then the question we must answer is, How much accuracy are we willing to sacrifice for the convenience of the lumped system analysis? Before answering this question, we should mention that a 15 percent uncertainty in the convection heat transfer coefficient h in most cases is considered “normal” and “expected.” Assuming h to be constant and uniform is also an approximation of questionable validity, especially for irregular geometries. Therefore, in the absence of sufficient experimental data for the specific geometry under consideration, we cannot claim our results to be better than 15 percent, even when Bi = 0. This being the case, introducing another source of uncertainty in the problem will not have much effect on the overall uncertainty, provided that it is minor. It is generally accepted that lumped system analysis is applicable if
Bi