Heat Transfer and Energy

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Heat Transfer and Energy

What is Heat?

Heat is Energy in Transit. Recall the First law from Thermodynamics.

∆

U = Q - W

What did we mean by all the terms?

• What is

∆

U ?

• What is Q ? • What is W ?

What is Heat Transfer?

Heat transfer is the transfer of Heat effected by a temperature difference. However, in contrast to what we may have done in

Thermodynamics, we are concerned with the rate of heat transfer. For that, we need a different energy balance, one that accounts for the rate of

transfer and production, a general energy balance. We will derive such a balance in later lectures.

Examples of Heat Transfer

• Pasteurizing milk

• Energy loss from a house by conduction through a window

• Melting of a polymer

• Cooling an automobile engine

• Cooling or heating a stream in a chemical process

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Lecture 1

Modes of Heat Transfer

• Conduction

• Convection • Radiation

Conduction

Conduction is the thermal equivalent of diffusion. In gases, energy is exchanged between molecules in collisions. You will recall from

Thermodynamics that a measure of the molecular motion and the energy exchange is the temperature and that conduction is the transfer of energy from the more energetic molecules to the less energetic. In the presence of a temperature gradient, energy is then transferred from the high temperature ( more energetic) to the lower temperature(less energetic). For liquids and solids, the example is not as clear. Conduction in liquids the collisions are more frequent and not as energetic and in solids transfer is by vibrations, for example, in a crystal lattice.

The constitutive equation describing the phenomenon is Fourier’s law

q = – k

∇

T

What we mean by the notation is the following:

q = q

_i

e

_i

_{T =}

∂

T

∂

x

i

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Conduction in one-dimension

To give you some notion of what I mean, examine a simple one-dimensional conduction problem.

q

_x

is the heat flux in watts/m

2

q

_x

= – k d T

d x

k is the thermal conductivity in

units of watts/m-°K

If we make an energy balance over a differential element at steady state,

q

_x x +∆x

– q

_x x

= 0

or q_x

= constant

Then, we see that

T = ax + b or dT

dx

=

T

₁

– T

₂

L

It follows that at steady state

q

_x

= – k

T

1

– T

2

L

Convection

Convection in heat transfer is a process which involves the diffusion of heat and the advection of energy by flow. The process can be described by Newton’s Law of Cooling

q

⋅

n = – h T – T

_b

where h is the heat transfer coefficient as watts/m2-°K

q

T

₁

T

₂

x

L

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Lecture 1

What is the relation between

h

expressed in English units and SI units ? In SI units , h = 10 watts/m2-°K

In English units, h = 1.7612 BTU/hr-ft2-°F

Quick estimates for conversion changes, divide watts/m2-°K by 6 to get BTU/hr-ft2-°F

Radiation

Thermal radiation is energy emitted by matter that is at a finite temperature. Emission occurs not only from solids, but from gases and liquids. The energy is carried by electromagnetic waves, originating at the expense of the internal energy of the matter. Conduction and convection depend on the presence of an intermediary. Radiation does not !

The normal component of heat flux emitted by a surface is given by the Stefan-Boltzmann’s Law

q

⋅

n =

εσ

T

_s4

where ε is the emissivity σ is the Stefan-Boltzman constant

The emissivity is the ratio of the energy emitted by the real surface compared to that emitted by an ideal surface (a black body). It is dimensionless.

σ = 5.67 x 10-8 watts/m2-K4 and T is in degrees Kelvin. The net rate of energy transfer between two surfaces

q

_i_→ _j

⋅

n

_j

= F

_ij

εσ

T

_i4

– T

4_j

The view factor, F_ij , depends on the distance R between and the orientation of the two surfaces.

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Problem Solving

Your textbook outlines a "stock" procedure for addressing, formulating and solving problems in Heat Transfer. In my view, it is a sound

technique and one that I would ask that you follow.

It involves several steps, each with a prescribed formula. What folloiws is the outline.

Problem Statement

The statement of the problem and the results required

Solution

1. Known

Read the problem carefully, then state briefly and concisely what is know about the problem. This is not a simple restaement of the

problem. 2. Find

Briefly and concisely state the results that are required.

3. Schematic

Draw a picture (schematic) of the system. Identify and label the heat transfer processes and the relevant boundaries of the system (control surfaces)

4. Assumptions

List all the relevant assumptions you will use in formulating the problem.

5. Properties

Compile the property valued for all the calculations to follow and identify the sources of the data.

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Lecture 1

Measurement of Thermal Conductivity

A Design Problem

Statement of the Problem

Measure the conductivity of the metal in a metal rod connected to a constant temperature sink (an ice bath)

Water flows through a well-mixed reservoir. In enters at Th1 and leaves at

the same temperature as it is in the water reservoir, Th2. The rod, the two

reservoirs are insulated from the surroundings.

Derive a relationship for the thermal conductivity and recommend a flow rate for the water.

Solution

1. Known

The heat lost by the fluid in the lower reservoir is transferred to the ice batch through the rod. The temperature is known for the ce bath. The water flow is measured and known, and we measure the water

temperatures in and out of the reservoir.

2. Find

Derive a relationship for the thermal conductivity and recommend a flow rate for the water.

3. Schematic

F

T

_h1

F

T

_h2

Ice bath

T

h 2 h 2

L = 6 in.

D = 0.5 in.

D

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4. Assumptions

There are a number of assumptions in this energy balance. What are they?

5. Properties

Some are given in the schematic, but what should we list ?

Data k (cal/s-cm-°K) k (Btu/h-ft-°F) Aluminum 0.45 108 Copper 0.9 216 Steel 0.13 32

Conditions

and Data

T_h1 = 80 °F T_c = 32 °F k = 32 BTU/ft-h-°F C_pw = 1 BTU/lb-°F = 62 lb./ft^3 L = 6 in. D = 0.5in.

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Lecture 1

6. Analysis

Energy balance on the rod

Heat exchanged from the water = heat transferred across the rod

Q =

ρ

_w

F

_w

C

_pw

T

_h1

– T

_h2

= k

L

π

D

2

4 T

h2

– T

c We did not state the constitutive relations . What are they?

There are a number of assumptions in this energy balance. What are they? We can solve for the thermal conductivity , k.. The result gives us an

equation by which we might better design the experiment.

k = 4L

π

D

θ

and the temperature rise can be expressed simply as

T

h1

– T

h2

T

_h1

– T

_c

=

θ

1 +

θ

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If we calculate the temperature rise as a function of flow rate we can obtain the following table.

In an experiment we can measure temperature no more precisely that 0.1 °F so that the highest flow rate would be 5 ml/min.

Some experimental questions

• How precise can you control and measure flow rates?

Flow rate