The Thermal Properties of Matter

To be able to use Fourier’s law, the thermal conductivity of the material must be known. This property, referred to as a transport property, gives an indication of the rate at which energy is transferred by the diffusion process and it depends on the physical structure of matter, atomic and molecular, which is related to the state of the matter.

Thermal Conductivity

From Fourier’s law (equation 6 from this blog entry), we can define the thermal conductivity associated with conduction in the x-direction as:

Analogous definitions are associated with thermal conductivities in the y- and z-directions (k_y, k_z), but for an isotropic material, the thermal conductivity is independent of the direction of transfer, k_x = k_y = k_z ≡ k.

Thus, for a given temperature gradient, the conduction heat flux increases with increasing thermal conductivity.

In general, the thermal conductivity of a solid is larger than that of a liquid, which is larger than that of a gas i.e., k_solid > k_liquid > k_gas. The thermal conductivity of a solid may be more than four orders of magnitude larger than that of a gas. This is largely due to differences in intermolecular spacing for the two states.

The Solid State: A solid may be comprised of free electrons and atoms bound in a periodic arrangement called a lattice. Therefore, transport of thermal energy may be due to two effects: the migration of free electrons and lattice vibrational waves (phonons). In pure metals, the electron contribution to conduction heat transfer dominates, whereas in nonconductors and semiconductors, the phonon contribution is dominant.

The Fluid State: This includes both liquids and gases.  The thermal energy transport is less effective in fluids due to the much larger intermolecular spacing and more random motion of molecules as compared to the solid state. Thus, the thermal conductivity of gases and liquids is generally smaller than that of solids.

The kinetic theory of gases can be used to explain the effect of temperature, pressure and chemical species on the thermal conductivity of a gas. From this theory, we know that thermal conductivity is directly proportional to the density of the gas,

From this theory it is known that the thermal conductivity is directly proportional to the density of the gas, the mean molecular speed c, and the mean free path λ_mfp, which is the average distance traveled by a molecule before experiencing a collision:

For an ideal gas, the mean free path may be expressed as:

Where: k_B is the Boltzmann’s constant, k_B = 1.381 x 10^-23 J/K and d is the diameter of the gas molecule.

As is expected, the mean free path is small for high pressure or low temperature due to the densely packed molecules. The mean free path also depends on the diameter of the molecule where larger molecules are more likely to experience collisions than small molecules; in the rare case of an infinitesimally small molecule, the molecules cannot collide, resulting in an infinite mean free path.

Other Relevant Properties

In the analysis of heat transfer problems, it is necessary to use several properties of matter. These properties are often referred to as thermophysical properties. These properties include two distinct categories:

• Transport Properties: includes the diffusion rate coefficients such as the thermal conductivity, k (for heat transfer), and the kinematic viscosity, ν (for momentum transfer).
• Thermodynamics Properties: These relate to the equilibrium state of a system. Examples include density (ρ) and specific heat (c_p). The volumetric heat capacity, ρc_p (J/m³K), measures the ability of a material to store thermal energy. Since substances, like solids and liquids, with large densities are characterized by small specific heats they are very good energy storage media while gases which have small densities are poor for thermal energy storage.

In heat transfer analysis, the ratio of the thermal conductivity to the heat capacity is an important property termed the thermal diffusivity α, with the units of m²/s:

Thermal diffusivity measures the ability of a material to conduct thermal energy relative to its ability to store thermal energy.

Materials of large α will respond quickly to changes in their thermal environment, while materials with small α will respond more slowly, taking longer to reach a new equilibrium condition.

The Conduction Rate Equation

The conduction rate equation, Fourier’s Law, was previously discussed in this blog entry. We can now consider its origin. Fourier’s Law is phenomenological, i.e., it is developed from observed phenomena rather than being derived from first principles, thus, we consider the rate equation as a generalization based on a lot of experimental evidence.

Let us consider the steady-state conduction experiment below where a cylindrical rod of known material is insulated on its lateral surface, while its end faces are maintained at different temperatures, where T1 > T2.

The temperature difference causes conduction heat transfer in the positive x-direction. We are able to measure the heat transfer rate q_x, and we seek to determine how q_x depends on the following variables:

• T, the temperature difference;
• x, the rod length;
• A, the cross-sectional area.

We can imagine first holding ΔT and Δx constant and varying A. Thus, we find that q_x is directly proportional to A. Similarly, holding ΔT and A constant, we see that q_x varies inversely with Δx. Finally, holding A and Δx constant, we find that q_x is directly proportional to ΔT. The collective effect is then:

Even if we change the material from a metal to a plastic, this proportionality will remain valid. But we would also find that, for equal values of A, Δx, and ΔT, the value of q_x would be smaller for the plastic than for the metal. This suggests that the proportionality may be converted to an equality by introducing a coefficient that is a measure of the material behavior. Thus:

where k, the thermal conductivity (W/m.K) is an important property of the material. Evaluating this expression in the limit as x → 0, we obtain the heat rate:

The minus sign is needed because heat is always transferred in the direction of decreasing temperature.

Fourier’s law, as written in Equation 2, suggests that the heat flux is a directional quantity. In particular, the direction of q_x′′ is normal to the cross-sectional area A. Or, more generally, the direction of heat flow will always be normal to a surface of constant temperature, called an isothermal surface.

The following figure shows the direction of heat flow q_x′′ in a plane wall for which the temperature gradient dT/dx is negative.

From Equation 2, it follows that q_x′′ is positive. Note that the isothermal surfaces are planes normal to the x-direction.

Recognizing that the heat flux is a vector quantity, we can write a more general statement of the conduction rate equation (Fourier’s law) as follows:

It can be understood through Equation 3 that the heat flux vector is in a direction perpendicular to the isothermal surfaces. An alternative form of Fourier’s law is therefore:

where q_n′′ is the heat flux in a direction n, which is normal to an isotherm, and n is the unit normal vector in that direction as shown below:

The heat transfer is sustained by a temperature gradient along n. Note also that the heat flux vector can be resolved into components such that, in Cartesian coordinates, the general expression for q′′ is:

Thus, from equation 3:

Each of the above expressions relates the heat flux across a surface to the temperature gradient in a direction perpendicular to the surface. It is also implied in Equation 3 that the medium in which the conduction occurs is isotropic. For such a medium, the value of the thermal conductivity is independent of the coordinate direction. 

Relationship Between Heat Transfer and the First Law of Thermodynamics

In this topic we are interested in the efficiency of heat engines. We are going to build upon the knowledge of thermodynamics and show how heat transfer plays an integral role in managing and promoting the efficiency of a wide range of energy conversion devices. 

Remember that we have defined a heat engine previously as any device that continuously or cyclically operates and converts heat to work. Power plants and thermoelectric devices are examples of heat engine. 

It is extremely important to improve the efficiency of heat engines. For example, an efficient combustion engine consumes less fuel to produce a given amount of work thus reduces emissions of pollutants. More efficient thermoelectric devices can generate more electricity from waste heat. 

The second law of thermodynamics can be represented in a number of distinct but comparable ways and is frequently employed when efficiency is an issue. The KelvinPlanck statement is very relevant to the operation of a heat engine. It states:

"It is impossible for any system to operate in a thermodynamic cycle and deliver a net amount of work to its surroundings while receiving energy by heat transfer from a single thermal reservoir"

Remember that a thermodynamic cycle is a process for which the initial and final states of the system are identical. Consequently, the energy stored in the system does not change between the initial and final states, and the first law of thermodynamics reduces to W = Q.

Because of the Kelvin–Planck statement, a heat engine must exchange heat with two or more reservoirs, gaining thermal energy from the higher-temperature reservoir and rejecting thermal energy to the lower-temperature reservoir. Therefore, converting all of the input heat to work is impossible and:

W = Q_in – Q_out,

where Q_in and Q_out are both defined to be positive. i.e.,

Q_in = heat transferred from the high temperature source to the heat engine

Q_out = the heat transferred from the heat engine to the low temperature sink.

The efficiency of a heat engine is the fraction of heat transferred into the system that is converted to work:

For a reversible process the ratio Q_out/Q_in is equal to the ratio of the absolute temperatures of the respective reservoirs (From the 2^nd Law of Thermodynamics. Thus, the efficiency of a heat engine undergoing a reversible process, i.e., Carnot efficiency, η_C, (as previously discussed) is given by:

where T_c and T_h are the absolute temperatures of the low and high temperature reservoirs, respectively. 

The Carnot efficiency is the maximum possible efficiency that any heat engine can achieve operating between those two temperatures. Any real heat engine, which will necessarily undergo an irreversible process, will have a lower efficiency.

From our knowledge of thermodynamics, we know that, for heat transfer to take place reversibly, it must occur through an infinitesimal temperature difference between the reservoir and heat engine. In heat transfer mechanisms, in order for heat transfer to occur, there must be a nonzero temperature difference between the reservoir and the heat engine. This introduces irreversibility and reduces the efficiency. 

Let us now consider a more realistic heat engine model where heat is transferred into the engine through a thermal resistance R_t,h, while heat is extracted from the engine through a second thermal resistance R_t,c where subscripts h and c refer to the hot and cold sides of the heat engine respectively. 

Let us now consider a more realistic heat engine model where heat is transferred into the engine through a thermal resistance R_t,h, while heat is extracted from the engine through a second thermal resistance R_t,c where subscripts h and c refer to the hot and cold sides of the heat engine respectively. This is shown in the following figure:

 

These thermal resistances are associated with heat transfer between the heat engine and the reservoirs across a nonzero temperature difference through mechanisms of conduction, convection and/or radiation. E.g., the resistances could represent conduction through walls separating the heat engine from the two reservoirs

Note that the reservoir temperatures are still T_h and T_c but that the temperatures seen by the heat engine are T_h,i < T_h and T_c,i > T_c, as shown in the diagram above. The heat engine is still assumed to be internally reversible, and its efficiency is still the Carnot efficiency.

However, the Carnot efficiency is now based on the internal temperatures T_h,i and T_c,i . Therefore in order to account for the realistic irreversible process, the efficiency, ηm, is as follows: 

where the ratio Q_out/Q_in, has been replaced by the corresponding ratio of heat rates, q_out/q_in. This replacement is based on applying energy conservation at an instant in time. Utilizing the definition of a thermal resistance, the heat transfer rates into and out of the heat engine are given by:

The above equations can be solved for the internal temperatures to result in:

For the T_c,i equation, q_out has already been related to q_in and η_m, The more realistic, modified efficiency can then be expressed as:

Solving for η_m, results in:

where R_tot = R_t,h + R_t,c.

It is evident that η_m = η_C only if the thermal resistances R_t,h and R_t,c could somehow be made infinitesimally small (or if qin  0). For realistic (nonzero) values of R_tot, η_m < η_C, and η_m further deteriorates as either R_tot or q_in increases. As an extreme case, note that η_m = 0 when T_h = T_c + q_inR_tot , meaning that no power could be produced even though the Carnot efficiency is nonzero.

In addition to the efficiency, another important parameter to consider is the power output of the heat engine, given by:

Relationship Between Heat Transfer and the First Law of Thermodynamics

• Thermodynamics and heat transfer are complementary engineering subjects, each addressing different aspects of energy behavior in systems.
• In thermodynamics, heat is treated as a form of energy transfer used to analyze system states and energy requirements, but the theory does not address how heat actually flows or at what rate.
• Heat transfer provides the engineering tools and rate equations needed to quantify heat-flow mechanisms (conduction, convection, radiation) and determine heat-exchange rates.
• Practical engineering design problems—such as sizing equipment, components, or entire systems—require heat-transfer analysis in addition to thermodynamic principles.
• Example: Designing and sizing a power plant cannot be accomplished using thermodynamics alone; engineers must apply heat-transfer principles to ensure feasible and efficient operation.

Relationship to the First Law of Thermodynamics (Conservation of Energy)

The first law of thermodynamics states that energy is conserved within a system. A system's energy can only change if energy crosses its boundaries.

For a closed system (fixed mass), there are only two mechanisms for energy transfer:

• Heat transfer across the system boundaries. 
• Work done by or on the system

These concepts form the standard mathematical statement of the first law for closed systems, as introduced in foundational thermodynamics courses:

Where ΔE_st^tot is the change in the total energy stored in the system, Q is the net heat transferred to the system, and W is the net work done by the system. This is schematically illustrated as follows: 

Figure 1

The first law of thermodynamics also applies to a control volume (open system), where mass can cross the system boundaries.

When mass enters or leaves a control volume, it carries energy with it — a process known as energy advection. Energy advection becomes a third mechanism for energy transfer in addition to heat transfer and work. Thus, for both closed systems and control volumes, the first law states that the change in system energy equals the net energy transferred across its boundaries.

First Law of Thermodynamics over a Time Interval (Δt)

A refresher for the 1st law of Thermodynamics:

1. The change in energy within a control volume is equal to the energy flowing into it minus the energy flowing out.

Energy can cross a control volume boundary through heat transfer, work, and energy advection (energy carried by mass flow). The first law of thermodynamics deals with total energy, which includes:

• Mechanical energy: kinetic + potential
• Internal energy: thermal energy plus chemical, nuclear, and other internal forms

In heat-transfer analysis, attention is mainly on thermal and mechanical energy. These two forms are not conserved on their own because they can be produced or consumed through conversions with other energy forms. Examples:

• Chemical reactions can decrease chemical energy and increase thermal energy.
• Electric motors convert electrical energy into mechanical energy.

These conversions can be viewed as thermal or mechanical energy generation (positive or negative). Therefore, a tailored form of the first law is needed for heat-transfer applications, accounting for these energy conversions.

Thermal and Mechanical Energy Equation over a Time Interval (Δt)

2. The rise in thermal and mechanical energy within a control volume equals the energy entering it, minus the energy leaving it, plus any thermal or mechanical energy produced inside the control volume.

This relation is written for a time period Δt, with all energy quantities expressed in joules. Because the first law must hold at every moment, it can also be written in terms of energy rates. In other words, at any instant, the rates of energy transfer must balance, with all terms expressed in joules per second (watts).

Thermal and Mechanical Energy Equation at an Instant (t)

3. The rate at which thermal and mechanical energy accumulates in a control volume equals the rate at which it enters, minus the rate at which it leaves, plus the rate at which it is produced inside the control volume.

If the combined inflow and generation of thermal and mechanical energy are greater than the outflow, the energy stored in the control volume will increase. If the outflow exceeds inflow and generation, the stored energy will decrease. When inflow and generation exactly match outflow, the system reaches a steady state, with no change in stored thermal and mechanical energy.

Let use now define the statement in italics and try to express it as an equation. let E stand for the sum of thermal and mechanical energy. Using the subscript st to represent energy stored in the control volume, the change in thermal and mechanical energy stored over the time interval Δt is then ΔE_st. The subscripts in and out refer to energy entering and leaving the control volume. Finally, thermal and mechanical energy generation is given the symbol E_g. Thus statement 1 is:

Statement 2 is represented as figure (b) in Figure 1 and is expressed as:

The above two equations are essential tools for solving heat transfer problems. Applying the first law begins with identifying an appropriate control volume and its control surface. The analysis typically follows a series of steps:

• the control surface is indicated, often by a dashed line;
• a decision is made whether to perform the analysis over a time interval Δt or on a rate basis, depending on the problem’s objectives and the form of the given data; finally,
• the relevant energy terms for the specific problem are identified.

The remainder of the section focuses on clarifying these energy terms to help develop confidence in applying them.

• Stored thermal and mechanical energy, E_st.
• Thermal and mechanical energy generation, E_g.
• Thermal and mechanical energy transport across the control surfaces, that is, the inflow and outflow terms, E_in and E_out.

the stored thermal and mechanical energy is given by:

E_st = Kinetic Energy (KE) + Potential Energy (PE) + thermal energy (Ut)

where Ut = Sensible Energy (U_sens) + Latent Enegry (U_lat) In many problems, the only relevant energy term will be the sensible energy, that is, E_st = U_sens.

The energy generation term is associated with conversion from some other form of internal energy (chemical, electrical, electromagnetic, or nuclear) to thermal or mechanical energy.

The inflow and outflow terms are surface phenomena. That is, they are associated exclusively with processes occurring at the control surface and are generally proportional to the surface area.

When the first law is applied to a control volume with fluid crossing its boundary, the work term is typically divided into two components. The first, called flow work, arises from pressure forces moving the fluid through the boundary and, for a unit mass, is equal to the product of the pressure and the fluid’s specific volume (pv). Under steady-state conditions (dEst/dt = 0), with no thermal or mechanical energy generation, the first law simplifies to the steady-flow energy equation. The equation for statement 2 becomes:

In most open system applications, changes in latent energy between the inflow and outflow conditions of the above equation may be neglected, thus the thermal energy reduces to only the sensible component.

If the fluid is approximated as an ideal gas with constant specific heats, the difference in enthalpies (per unit mass) between the inlet and outlet flows may then be expressed as:

(i_in  i_out) = c_p(T_in - T_out)

Where:

• c_p is the specific heat at constant pressure
• T_in and T_out are the inlet and outlet temperatures, respectively.
• If the fluid is an incompressible liquid, its specific heats at constant pressure and volume are equal, c_p = c_v = c,

Thus, for the above equation, the change in sensible energy (per unit mass) reduces to

(u_t,in - u_t,out) = c(T_in - T_out).

Unless the pressure drop is extremely large, the difference in flow work terms, (pv)_in = (pv)_out, is negligible for a liquid.

Having already assumed steady-state conditions, no changes in latent energy, and no thermal or mechanical energy generation, the simplified steady-flow thermal energy equation is obtained as follows:

Heat Exchanger

In this blog entry we will look into the thermodynamics of a heat exchanger. To learn more about types of heat exchangers and factors affecting performance of a heat exchanger, you can read a previous blog entry on Heat Exchangers here.

To recap, a heat exchanger is a device with two flowing streams that exchange heat without mixing. The most basic form of a heat exchanger is a double-pipe heat exchanger that consists of two concentric pipes with differing diameters. One of the fluids flows in the inner pipe while the other flows in the annular space between the two pipes. The two fluid should have a considerable temperature difference to facilitate heat transfer from hot to cold fluid through the pipe walls. 

The figure below illustrates the layout of co-current and countercurrent flow heat exchangers:

No work is produced in a heat exchanger and both kinetic and potential energy changes are negligible. The energy equation considering a heat exchanger as a system thus reduces to:

The outer shell of the heat exchanger is usually well insulated to prevent heat loss to the surroundings.

Note: For the following example, Appendices B.2 and B.4 for steam values that I have referred to in these questions was obtained from: 

M.D Koretsky, Engineering and Chemical Thermodynamics, Wiley, 2004.

Example

High pressure steam at 0.4 kg/s and 2 MPa and 450°C enters an adiabatic steady flow turbine. The work produced by the turbine is 400 hp. The exit stream from the turbine is at 40 kPa and it is fed to a heat exchanger where it is condensed at constant pressure to obtain saturated liquid. As cooling medium liquid water is used which enters the heat exchanger at 15°C and leaves at 55°C. Assume no heat losses from the heat exchanger to the surroundings.

a) What is the condition of the steam leaving the turbine?

b) Calculate the mass flow rate of the cooling water used

Solution 1

Solution 2

An alternate solution can be as follows:

Choosing the turbine and the heat exchanger as the system, the energy equation can be written and solved as follows:

Steady-State PFR with Heat Exchange

 

CSTR - Non-Adiabatic Case

The Effect of Reaction Heat on a Non-Adiabatic CSTR