# Etiket Arşivleri: Heat Transfer

## Heat Transfer by Convection and Radiation

**FE 546 Thermal Process Engineering**

# Heat Transfer

Heat Transfer by Convection and Radiation

Heat Transfer by Convection

Convective heat transfer inside containers results either from the natural effects of changes in density in the liquid induced by changes in temperature at the container walls (free or natural convection) or by creating motion in the container contents by axial or end-over-end rotation (forced convection).

# Mechanism of Natural Convection

# The process of natural convection initially involves heat transferred by conduction into the outer layers of fluid adjacent to the heated wall; this results in a decrease in the density, and the heated fluid layer rises.

When it reaches the top of the liquid at the headspace, the induced fluid motion causes it to fall in the central core, the displaced hot fluid at the wall being replaced by colder fluid in the core.

As the temperature of the can contents becomes more uniform and the driving force smaller, the fluid velocity tends to decrease and eventually, when the fluid becomes uniformly heated, the motion ceases. Idealized representation of convection currents in a can of liquid

In the upper part of the container the hot liquid is being pumped by the heated fluid rising in the boundary layer, and being placed on the cold liquid in the core.

Simultaneously, in the case of a vertical container, there is also heat rising from the bottom end of the can, which produces mixing eddies in the bulk of the fluid.

Datta (1985) has shown that as result of instabilities in the temperature distribution, regular bursts of hot liquid occur on the base of the container. This phenomenon is known as Bernard convection.

The mechanism of unsteady-state convection is very complex and varies with time of heating and/or cooling; consequently it is very difficult to model precisely.

Hiddink (1975) used a flow visualizing technique with metallic powders in liquids in containers with light- transmitting walls, to highlight the streamlines in the bulk of the fluid.

Blaisdell (1963), have used thermocouples at several points in the containers to plot the temperature profiles.

Guerrei (1993) has discussed the internal flow patterns in convection-heating bottles of cylindrical and square shape, using the concept of a double-layer system of rising hot liquid on the wall of the container which discharge into a central volume. The point-of- slowest-heating was shown to be 0.006 m from the wall of an 0.06 m internal diameter container.

The most important point, from a practical safety point of view, is where the slowest heating point is situated.

While the discussion has been concerned with convection heating/cooling inside containers, there is also the problem of convection heating on the outside, from the heating or cooling medium to the container wall.

In the case of pure steam heating, condensation of steam on the container wall surface raises the temperature of the surface almost immediately to that of the steam and, consequently, no problem arises.

If, however, the steam contains air, either as an adulterant or intentionally, then the velocity of the mixture over the surface will affect the temperature of the surface and a more complex situation will arise.

Similarly, if water is used for heating, and also for cooling, then the wall temperature will be affected by the velocity of the heat transfer medium.

In all cases, except for condensing pure steam, it is necessary to consider external convective heat transfer to the outer surface of the container.

# Basic Concepts in Convection Heat Transfer

# Mathematical models for the prediction of temperatures in the heating of canned foods by convection are necessary in order to determine the process requirements.

There are three approaches to convective heat transfer:

1) the film theory

2) the use of dimensionless numbers

3) the more rigorous mathematical treatment of the basic fluid dynamic and heat transfer models.

# 1) Film Theory

# The idealized temperature profile for a material being heated by a fluid and separated by a container wall Thus if T1 is the temperature of the heating medium and T4 is the temperature in the bulk of the fluid being heated, and the average film temperatures are T2 and T3, then an overall heat-transfer coefficient U (alternatively denoted by H) can be defined as follows:

# 2) Correlations for Predicting Heat-Transfer Coefficients

# Engineering practice makes use of dimensionless numbers for correlating heat transfer coefficients with the physical circumstances of heat transfer and the physical properties and flow conditions of the fluids involved.

The four dimensionless numbers used in heat transfer studies are:

For forced convection; the correlation is

# 3) Models for Convection Heat Transfer

# Some of the models that have been used to predict temperature distributions and velocity profiles in heated and cooled can liquid products.

The models may be classified as follows:

²Energy Balance Model

This is the simplest of the models.

By equating the overall rate of heat transfer into the

can with the rate of accumulation of heat inside the can, an energy balance equation can be established.

This equation shows that the rate of heating is an exponential function, which depends on the overall heat-transfer coefficient U (or the internal heat-transfer coefficient hint when steam is used as the heating medium with metallic cans), the surface area A, the mass of the contents and their specific heat, as well as the temperature of the heating medium and the initial temperature of the contents T0.

²Effective Thermal Diffusivity Model

This model makes use of the unsteady state

conduction model solutions and an apparent or effective thermal diffusivity.

This depended on the ratio of solids to liquid in the

container.

²Transport Equation Model

This is the most rigorous approach to determining the temperature distributions and the velocity profiles in containers filled with liquids.

The form of the equations depends on the geometry of the container; care should be taken to use the most appropriate coordinate system.

The equations which have to be solved in relation to the container boundaries are the equation of continuity:

# Radiation Heating

# Thermal radiation has a wavelength from 0.8 to 400μm and requires no medium to transmit its energy.

The transfer of energy by radiation involves three processes:

1)the conversion of the thermal energy of a hot source

into electromagnetic waves

2) the passage of the waves from the hot source to the cold receiver

3) the absorption and reconversion of the electromagnetic waves into thermal energy.

The quantity of energy radiated from the surface per unit time is the emissive power E of the surface.

For a perfect radiator, known as a black body, the emissive power is proportional to the fourth power of the temperature T :

The main application for radiation theory is in

relation to can sterilization using gaseous flames. The cooling of cans to the atmosphere also involves loss of heat by radiation as well as convection.

A more complex heat-transfer model for canned foods heated in a flame sterilizer was developed.

6. Radiation exchange only occurs between the flame and the can, and from the can to the surroundings.

7. The temperature of the flame is the adiabatic temperature calculated on the dissociation of the combustion species.

8. The dissociated species do not combine on the can surface.

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## Heat Transfer Lecture Notes ( Dr. A. Rıza TEKİN )

**FE321 Heat Transfer Lecture Notes**

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## Heat Transfer Lecture Slides II ( Dr. Gregory A. Kallio )

# ME 259

Heat Transfer

Lecture Slides II

Dr. Gregory A. Kallio

Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology

California State University, Chico

Steady-State Conduction Heat Transfer

Incropera & DeWitt coverage:

–Chapter 2: General Concepts of Heat Conduction

–Chapter 3: One-Dimensional, Steady-State Conduction

–Chapter 4: Two-Dimensional, Steady-State Conduction

General Concepts of Heat Conduction

Reading: Incropera & DeWitt

Chapter 2

Generalized Heat Conduction

Fourier’s law, 1-D form:

Fourier’s law, general form:

-q” is the heat flux vector, which has three components; in Cartesian coordinates:

(magnitude)

The Temperature Gradient

ÑT is the temperature gradient, which is:

–a vector quantity that points in direction of maximum temperature increase

–always perpendicular to constant temperature surfaces, or isotherms

(Cartesian)

(Cylindrical)

(Spherical)

Thermal Conductivity

k is the thermal conductivity of the material undergoing conduction, which is a tensor quantity in the most general case:

–most materials are homogeneous, isotropic, and their structure is time-independent; hence:

which is a scalar and usually assumed to be a constant if evaluated at the average temperature of the material

Total Heat Rate

Total heat rate (q) is found by integrating the heat flux over the appropriate area:

k and Ñ T must be known in order to calculate q” from Fourier’s law

–k is usually obtained from material property tables

–to find ÑT, another equation is required; this additional equation is derived by applying the conservation of energy principle to a differential control volume undergoing conduction heat transfer; this yields the general Heat Diffusion (Conduction) Equation

Heat Diffusion (Conduction) Equation

For a homogeneous, isotropic solid material undergoing heat conduction:

Cylindrical and spherical coordinate system forms given in text (p. 64-65)

This is a second-order, partial differential equation (PDE); its solution yields the temperature field, T(x,y,z,t), within a given solid material

Heat Diffusion (Conduction) Equation

For constant thermal conductivity (k):

For k = constant, steady-state conditions, and no internal heat generation

–this is known as Laplace’s equation, which appears in other branches of engineering science (e.g., fluids, electrostatics, and solid mechanics)

Boundary Conditions and Initial Condition

Boundary Conditions: known conditions at solution domain boundaries

Initial Condition: known condition at t = 0

Number of boundary conditions required to solve the heat diffusion equation is equal to the number of spatial dimensions multiplied by two

There is only one initial condition, which takes the form

–where Ti may be a constant or a function of x,y, and z

Types of Boundary Conditions for Conduction Problems

Specified surface temperature, e.g.,

Specified surface heat flux, e.g.,

Specified convection (h, T¥ given), e.g.,

Specified radiation (e, Tsur given), e.g.,

Solving the Heat Diffusion Equation

Choose a coordinate system that best fits the problem geometry.

Identify the independent variables (x,y,z,t), e,g, is it a S-S problem? Is conduction 1-D, 2-D, or 3-D? Justify assumptions.

Determine if k can be treated as constant and if

Write the general heat conduction equation using the chosen coordinates.

Reduce equation to simplest form based upon assumptions.

Write boundary conditions and initial condition (if applicable).

Obtain a general solution for T(x,y,z,t) by some method; if impossible, resort to numerical methods.

Solving the Heat Diffusion Equation, cont.

Solve for the constants in the general solution by applying the boundary conditions and initial condition to obtain a particular solution.

Check solution for correctness (e.g., at boundaries or limits such as x = 0, t = 0, t ® ¥ , etc.)

Calculate heat flux or total heat rate using Fourier’s law, if required.

Optional: rearrange solution into a nondimensional form

Example:

GIVEN: Rectangular copper bar of dimensions L x W x H is insulated on the bottom and initially at Ti throughout . Suddenly, the ends are subjected and maintained at temperatures T1 and T2 , respectively, and the other three sides are exposed to forced convection with known h, T¥.

FIND: Governing heat equation, BCs, and initial condition

One-Dimensional, Steady-State Heat Conduction

Reading: Incropera & DeWitt,

Chapter 3

1-D, S-S Conduction in Simple Geometries w/o Heat Generation

Plane Wall

–if k = constant, general heat diffusion equation reduces to

–separating variables and integrating yields

–where T(x) is the general solution; C1 and C2 are integration constants that are determined from boundary conditions

1-D, S-S Conduction in Simple Geometries w/o Heat Generation

Plane Wall, cont.

–suppose the boundary conditions are

–integration constants are then found to be

–the particular solution for the temperature distribution in the plane wall is now

1-D, S-S Conduction in Simple Geometries w/o Heat Generation

Plane wall, cont.

–The conduction heat rate is found from Fourier’s law:

–If k were not constant, e.g., k = k(T), the analysis would yield

»note that the temperature distribution would be nonlinear, in general

1-D, S-S Conduction in Simple Geometries w/o Heat Generation

Electric Circuit Analogy

–heat rate in plane wall can be written as

–in electrical circuits we have Ohm’s law:

–analogy:

Thermal Circuits for Plane Walls

Series Systems

Parallel Systems

Thermal Circuits for Plane Walls, cont.

Complex Systems

Thermal Resistances for Other Geometries Due to Conduction

Cylindrical Wall

Spherical Wall

Convective & Radiative Thermal Resistance

Convection

Radiation

Critical Radius Concept

Since the surface areas of cylinders and spheres increase with r, there exist competing heat transfer effects with the addition of insulation under convective boundary conditions (see Example 3.4)

A critical radius (rcr) exists for radial systems, where:

–adding insulation up to this radius will increase heat transfer

– adding insulation beyond this radius will decrease heat transfer

For cylindrical systems, rcr = kins/h

For spherical systems, rcr = 2kins/h

Thermal Contact Resistance

Thermal contact resistance exists at solid-solid interfaces due to surface roughness, creating gaps of air or other material:

Thermal Contact Resistance

R”t,c is usually experimentally measured and depends upon

–thermal conductivity of solids A and B

–surface finish & cleanliness

–contact pressure

–gap material

–temperature at contact plane

See Tables 3.1, 3.2 for typical values

EXAMPLE

Given: two, 1cm thick plates of milled, cold-rolled steel, 3.18mm roughness, clean, in air under 1 MPa contact pressure

Find: Thermal circuit and compare thermal resistances

1-D, S-S Conduction in Simple Geometries with Heat Generation

Thermal energy can be generated within a material due to conversion from some other energy form:

–Electrical

–Nuclear

–Chemical

Governing heat diffusion equation if k = constant:

S-S Heat Transfer from Extended Surfaces (i.e., fins)

Consider plane wall exposed to convection where Ts>T¥:

How could you enhance q ?

–increase h

–decrease T¥

–increase As (attach fins)

Fin Nomenclature

x = longitudinal direction of fin

L = fin length (base to tip)

Lc = fin length corrected for tip area

W = fin width (parallel to base)

t = fin thickness at base

Af = fin surface area exposed to fluid

Ac = fin cross-sectional area, normal to heat flow

Ap = fin (side) profile area

P = fin perimeter that encompasses Ac

D = pin fin diameter

Tb = temperature at base of fin

1-D Conduction Model for Thin Fins

If L >> t and k/L >> h, then the temperature gradient in the longitudinal direction (x) is much greater than that in the transverse direction (y); therefore

Another way of viewing fin heat transfer is to imagine 1-D conduction with a negative heat generation rate along its length due to convection

Fin Performance

Fin Effectiveness

Fin Efficiency

–for a straight fin of uniform cross-section:

–where Lc = L + t / 2 (corrected fin length)

Calculating Single Fin Heat Rate from Fin Efficiency

Calculate corrected fin length, Lc

Calculate profile area, Ap

Evaluate parameter

# Determine fin efficiency hf from Figure 3.18, 3.19, or Table 3.5

Calculate maximum heat transfer rate from fin:

Calculate actual heat rate:

Maximum Heat Rate for Fins of Given Volume

Analysis:

“Optimal” design results:

Fin Thermal Resistance

Fin heat rate:

Define fin thermal resistance:

Single fin thermal circuit:

Analysis of Fin Arrays

Total heat transfer =

heat transfer from N fins +

heat transfer from exposed base

Thermal circuit:

–where

Analysis of Fin Arrays, cont.

Overall thermal resistance:

Example

Given: Annular array of 10 aluminum fins, spaced 4mm apart C-C, with inner and outer radii of 1.35 and 2.6 cm, and thickness of 1 mm. Temperature difference between base and ambient air is 180°C with a convection coefficient of 125 W/m2-K. Contact resistance of 2.75×10-4 m2-K/W exists at base.

Find: a) Total heat rate w/o and with fins

b) Effect of R”t,c on heat rate

Two-Dimensional, Steady-State Heat Conduction

Reading: Incropera & DeWitt

Chapter 4

Governing Equation

Heat Diffusion Equation reduces to:

Solving the HDE for 2-D, S-S heat conduction by exact analysis is impossible for all but the most simple geometries with simple boundary conditions.

Solution Methods

Analytical Methods

–Separation of variables (see section 4.2)

–Laplace transform

–Similarity technique

–Conformal mapping

Graphical Methods

–Plot isotherms & heat flux lines

Numerical Methods

–Finite-difference method (FDM)

–Finite-element method (FEM)

Conduction Shape Factor

The heat rate in some 2-D geometries that contain two isothermal boundaries (T1, T2) with k = constant can be expressed as

–where S = conduction shape factor

(see Table 4.1)

Define 2-D thermal resistance:

Conduction Shape Factor, cont.

Practical applications:

–Heat loss from underground spherical tanks: Case 1

–Heat loss from underground pipes and cables: Case 2, Case 4

–Heat loss from an edge or corner of an object: Case 8, Case 9

–Heat loss from electronic components mounted on a thick substrate: Case 10

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## Heat Transfer Past Exams-1

# FE 321 Past Exams – Gaziantep University

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## Heat Transfer Past Exams-2

# FE 321 Heat Transfer

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## Heat Transfer / Heat Exchanger

# How is the heat transfer?

# Mechanism of Convection

# Applications

# Mean fluid Velocity and Boundary and their effect on the rate of heat transfer.

# Fundamental equation of heat transfer

# Logarithmic-mean temperature difference

# Heat transfer Coefficients

# Heat flux and Nusselt correlation

# Simulation program for Heat Exchanger

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## Introduction to Engineering Heat Transfer

# These notes provide an introduction to engineering heat transfer. Heat transfer processes set limits to the performance of aerospace components and systems and the subject is one of an enormous range of application. The notes are intended to describe the three types of heat transfer and provide basic tools to enable the readers to estimate the magnitude of heat transfer rates in realistic aerospace applications. There are also a number of excellent texts on the subject; some accessible references which expand the discussion in the notes are listen in the bibliography.

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