In describing heat transfer problems, students often make the mistake of interchangeably using the terms heat and temperature. Actually, there is a distinct difference between the two. Temperature is a measure of the amount of energy possessed by the molecules of a substance. It is a relative measure of how hot or cold a substance is and can be used to predict the direction of heat transfer. The symbol for temperature is T. The common scales for measuring temperature are the Fahrenheit, Rankine, Celsius, and Kelvin temperature scales. Heat is energy in transit. The transfer of energy as heat occurs at the molecular level as a result of a temperature difference. Heat is capable of being transmitted through solids and fluids by conduction, through fluids by convection, and through empty space by radiation. The symbol for heat is Q. Common units for measuring heat are the British Thermal Unit (Btu) in the English system of units and the calorie in the SI system (International System of Units). 

HEAT AND WORK 

Distinction should also be made between the energy terms heat and work. Both represent energy in transition. Work is the transfer of energy resulting from a force acting through a distance. Heat is energy transferred as the result of a temperature difference. Neither heat nor work are thermodynamic properties of a system. Heat can be transferred into or out of a system and work can be done on or by a system, but a system cannot contain or store either heat or work. Heat into a system and work out of a system are considered positive quantities. When a temperature difference exists across a boundary, the Second Law of Thermodynamics indicates the natural flow of energy is from the hotter body to the colder body. The Second Law of Thermodynamics denies the possibility of ever completely converting into work all the heat supplied to a system operating in a cycle. The Second Law of Thermodynamics, described by Max Planck in 1903, states that: It is impossible to construct an engine that will work in a complete cycle and produce no other effect except the raising of a weight and the cooling of a reservoir. The second law says that if you draw heat from a reservoir to raise a weight, lowering the weight will not generate enough heat to return the reservoir to its original temperature, and eventually the cycle will stop. If two blocks of metal at different temperatures are thermally insulated from their surroundings and are brought into contact with each other the heat will flow from the hotter to the colder. Eventually the two blocks will reach the same temperature, and heat transfer will cease. Energy has not been lost, but instead some energy has been transferred from one block to another.

Modes of Transferring Heat

Heat is always transferred when a temperature difference exists between two bodies. There are three basic modes of heat transfer:

Conduction involves the transfer of heat by the interactions of atoms or molecules of a material through which the heat is being transferred.

Convection involves the transfer of heat by the mixing and motion of macroscopic portions of a fluid.

Radiation, or radiant heat transfer, involves the transfer of heat by electromagnetic radiation that arises due to the temperature of a body.

The three modes of heat transfer will be discussed in greater detail 

HEAT FLUX 

The rate at which heat is transferred is represented by the symbol Q. Common units for heat transfer rate is Btu/hr. Sometimes it is important to determine the heat transfer rate per unit area, or heat flux, which has the symbol Q” . Units for heat flux are Btu/hr-ft^2. The heat flux can be determined by dividing the heat transfer rate by the area through which the heat is being transferred. 

Q” = Q / A

where:

          Q”= heat flux (Btu/hr-ft^2)

Q = heat transfer rate (Btu/hr)

          A = area (ft^2) 

THERMAL CONDUCTIVITY 

The heat transfer characteristics of a solid material are measured by a property called the thermal conductivity (k) measured in Btu/hr-ft-ºF. It is a measure of a substance’s ability to transfer heat through a solid by conduction. The thermal conductivity of most liquids and solids varies with temperature. For vapors, it depends upon pressure. 

LOG MEAN TEMPERATURE DIFFERENCE 

In heat exchanger applications, the inlet and outlet temperatures are commonly specified based on the fluid in the tubes. The temperature change that takes place across the heat exchanger from the entrance to the exit is not linear. A precise temperature change between two fluids across the heat exchanger is best represented by the log mean temperature difference (LMTD),

 LMTD = (∆T2 - ∆T1) / ln (∆T2 /∆T1) 

Where: 

∆T2 = the larger temperature difference between the two fluid streams at either the entrance or the exit to the heat exchanger

∆T1 = the smaller temperature difference between the two fluid streams at  either the entrance or the exit to the heat exchanger 

CONVECTIVE HEAT TRANSFER COEFFICIENT 

The convective heat transfer coefficient (h), defines, in part, the heat transfer due to convection. The convective heat transfer coefficient is sometimes referred to as a film coefficient and represents the thermal resistance of a relatively stagnant layer of fluid between a heat transfer surface and the fluid medium. Common units used to measure the convective heat transfer coefficient are Btu/hr – ft^2 - ºF. 

OVERALL HEAT TRANSFER COEFFICIENT 

In the case of combined heat transfer, it is common practice to relate the total rate of heat transfer (Q), the overall cross-sectional area for heat transfer (Ao), and the overall temperature difference (∆To) using the overall heat transfer coefficient (Uo). The overall heat transfer coefficient combines the heat transfer coefficient of the two heat exchanger fluids and the thermal conductivity of the heat exchanger tubes. Uo is specific to the heat exchanger and the fluids that are used in the heat exchanger. 

Q = Uo Ao ∆To 

where: 

 Q    = the rate heat of transfer (Btu/hr) 

Uo   = the overall heat transfer coefficient (Btu/hr – ft^2 - ºF)

Ao   = the overall cross-sectional area for heat transfer (ft^2)

∆To = the overall temperature difference (ºF)

CONDUCTION HEAT TRANSFER

Conduction heat transfer is the transfer of thermal energy by interactions between adjacent atoms and molecules of a solid. 

Conduction

Conduction involves the transfer of heat by the interaction between adjacent molecules of a material. Heat transfer by conduction is dependent upon the driving "force" of temperature difference and the resistance to heat transfer. The resistance to heat transfer is dependent upon the nature and dimensions of the heat transfer medium. All heat transfer problems involve the temperature difference, the geometry, and the physical properties of the object being studied. In conduction heat transfer problems, the object being studied is usually a solid. Convection problems involve a fluid medium. Radiation heat transfer problems involve either solid or fluid surfaces, separated by a gas, vapor, or vacuum. There are several ways to correlate the geometry, physical properties, and temperature difference of an object with the rate of heat transfer through the object. In conduction heat transfer, the most common means of correlation is through Fourier’s Law of Conduction. The law, in its equation form, is used most often in its rectangular or cylindrical form (pipes and cylinders), both of which are presented below. 

Rectangular  Q =  k A (∆T / ∆x) 

Cylindrical    Q =  k A (∆T / ∆r) 

where:

   Q     = rate of heat transfer (Btu/hr) 

   A     = cross-sectional area of heat transfer (ft^2)

   ∆x   = thickness of slab (ft)

   ∆r    = thickness of cylindrical wall (ft)

   ∆T   = temperature difference (°F)

   k     = thermal conductivity of slab (Btu/ft-hr-°F) 

CONVECTION HEAT TRANSFER

Heat transfer by the motion and mixing of the molecules of a liquid or gas is called convection.

 Convection

Convection involves the transfer of heat by the motion and mixing of "macroscopic" portions of a fluid (that is, the flow of a fluid past a solid boundary). The term natural convection is used if this motion and mixing is caused by density variations resulting from temperature differences within the fluid. The term forced convection is used if this motion and mixing is caused by an outside force, such as a pump. The transfer of heat from a hot water radiator to a room is an example of heat transfer by natural convection. The transfer of heat from the surface of a heat exchanger to the bulk of a fluid being pumped through the heat exchanger is an example of forced convection.

Heat transfer by convection is more difficult to analyze than heat transfer by conduction because no single property of the heat transfer medium, such as thermal conductivity, can be defined to describe the mechanism. Heat transfer by convection varies from situation to situation (upon the fluid flow conditions), and it is frequently coupled with the mode of fluid flow. In practice, analysis of heat transfer by convection is treated empirically (by direct observation). Convection heat transfer is treated empirically because of the factors that affect the stagnant film thickness:

1. Fluid velocity

2. Fluid viscosity

3. Heat flux

4. Surface roughness

5. Type of flow (single-phase/two-phase) 

Convection involves the transfer of heat between a surface at a given temperature (Ts) and fluid at a bulk temperature (Tb). The exact definition of the bulk temperature (Tb) varies depending on the details of the situation. For flow adjacent to a hot or cold surface, Tb is the temperature of the fluid "far" from the surface. For boiling or condensation, Tb is the saturation temperature of the fluid. For flow in a pipe, Tb is the average temperature measured at a particular cross section of the pipe. 

The basic relationship for heat transfer by convection has the same form as that for heat transfer by conduction:

 Q = h A ∆T

where:

 Q   = rate of heat transfer (Btu/hr)

 h   = convective heat transfer coefficient (Btu/hr-ft^2-°F)

 A   = surface area for heat transfer (ft^2)

 ∆T = temperature difference (°F) 

The convective heat transfer coefficient (h) is dependent upon the physical properties of the fluid and the physical situation. Typically, the convective heat transfer coefficient for laminar flow is relatively low compared to the convective heat transfer coefficient for turbulent flow. This is due to turbulent flow having a thinner stagnant fluid film layer on the heat transfer surface. Values of h have been measured and tabulated for the commonly encountered fluids and flow situations occurring during heat transfer by convection. 

RADIANT HEAT TRANSFER

Radiant heat transfer is thermal energy transferred by means of electromagnetic waves or particles. 

Thermal Radiation

Radiant heat transfer involves the transfer of heat by electromagnetic radiation that arises due to the temperature of a body. Most energy of this type is in the infra-red region of the electromagnetic spectrum although some of it is in the visible region. The term thermal radiation is frequently used to distinguish this form of electromagnetic radiation from other forms, such as radio waves, x-rays, or gamma rays. The transfer of heat from a fireplace across a room in the line of sight is an example of radiant heat transfer. Radiant heat transfer does not need a medium, such as air or metal, to take place. Any material that has a temperature above absolute zero gives off some radiant energy. When a cloud covers the sun, both its heat and light diminish. This is one of the most familiar examples of heat transfer by thermal radiation. 

Black Body Radiation

A body that emits the maximum amount of heat for its absolute temperature is called a black body. Radiant heat transfer rate from a black body to its surroundings can be expressed by the following equation. 

 Q = б AT^4 

where:

Q = heat transfer rate (Btu/hr)

б = Stefan-Boltzman constant (0.174 Btu/hr-ft^2-°R^4)

A = surface area (ft^2)

T = temperature (°R) 

Two black bodies that radiate toward each other have a net heat flux between them. The net flow rate of heat between them is given by an adaptation of Equation. 

Q = б A (T1^4 – T2^4 ) 

where:

A = surface area of the first body (ft^2)

T1 = temperature of the first body (°R)

T2 = temperature of the second body (°R) 

All bodies above absolute zero temperature radiate some heat. The sun and earth both radiate heat toward each other. This seems to violate the Second Law of Thermodynamics, which states that heat cannot flow from a cold body to a hot body. The paradox is resolved by the fact that each body must be in direct line of sight of the other to receive radiation from it. Therefore, whenever the cool body is radiating heat to the hot body, the hot body must also be radiating heat to the cool body. Since the hot body radiates more heat (due to its higher temperature) than the cold body, the net flow of heat is from hot to cold, and the second law is still satisfied.

 Emissivity

Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and are called gray bodies. To take into account the fact that real objects are gray bodies, Equation is modified to be of the following form. 

Q = Є б A T^4

 where: 

Є = emissivity of the gray body (dimensionless) 

Emissivity is simply a factor by which we multiply the black body heat transfer to take into account that the black body is the ideal case. Emissivity is a dimensionless number and has a maximum value of 1.0. 

Radiation Configuration Factor

Radiative heat transfer rate between two gray bodies can be calculated by the equation stated below.

Q = fa fe б A (T1^4 – T2^4) 

where: 

fa = is the shape factor, which depends on the spatial arrangement of the two objects (dimensionless)

fe = is the emissivity factor, which depends on the emissivities of both objects (dimensionless)

The two separate terms fa and fe can be combined and given the symbol f. The heat flow between two gray bodies can now be determined by the following equation: 

Q =  f б A (T1^4 – T2^4) 

The symbol (f) is a dimensionless factor sometimes called the radiation configuration factor, which takes into account the emissivity of both bodies and their relative geometry. The radiation configuration factor is usually found in a text book for the given situation. Once the

configuration factor is obtained, the overall net heat flux can be determined. Radiant heat flux should only be included in a problem when it is greater than 20% of the problem. 

Heat Transfer Terminology Summary 

ü       Heat is energy transferred as a result of a temperature difference. 

ü       Temperature is a measure of the amount of molecular energy   contained in a substance. 

ü       Work is a transfer of energy resulting from a force acting through a distance. 

ü       The Second Law of Thermodynamics implies that heat will not transfer from a colder to a hotter body without some external source of energy. 

ü       Conduction involves the transfer of heat by the interactions of atoms or molecules of a material through which the heat is being transferred. 

ü       Convection involves the transfer of heat by the mixing and motion of macroscopic portions of a fluid. 

ü       Radiation, or radiant heat transfer, involves the transfer of heat by electromagnetic radiation that arises due to the temperature of a body. 

ü       Heat flux is the rate of heat transfer per unit area. 

ü       Thermal conductivity is a measure of a substance’s ability to transfer heat through itself. 

ü       Log mean temperature difference is the ∆T that most accurately represents the ∆T for a heat exchanger. 

ü       The local heat transfer coefficient represents a measure of the ability to transfer heat through a stagnant film layer. 

ü       The overall heat transfer coefficient is the measure of the ability of a heat exchanger to transfer heat from one fluid to another. 

ü       The bulk temperature is the temperature of the fluid that best represents the majority of the fluid which is not physically connected to the heat transfer site. 

Conduction Heat Transfer Summary 

ü       Conduction heat transfer is the transfer of thermal energy by interactions between adjacent molecules of a material. 

ü       Fourier’s Law of Conduction can be used to solve for rectangular and cylindrical coordinate problems. 

ü       Heat flux (Q” ) is the heat transfer rate (Q ) divided by the area (A).  

ü       Heat conductance problems can be solved using equivalent    resistance formulas analogous to electrical circuit problems. 

Convection Heat Transfer Summary 

ü       Convection heat transfer is the transfer of thermal energy by the mixing and motion of a fluid or gas. 

ü       Whether convection is natural or forced is determined by how the medium is placed into motion. 

ü       When both convection and conduction heat transfer occurs, the overall heat transfer coefficient must be used to solve problems. 

ü       The heat transfer equation for convection heat transfer is Q= h A ∆T 

Radiant Heat Transfer Summary 

ü       Black body radiation is the maximum amount of heat that can be transferred from an ideal object. 

ü       Emissivity is a measure of the departure of a body from the ideal black body. 

ü       Radiation configuration factor takes into account the emittance and relative geometry of two objects.