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:

Maxwell Equation Solved Example

Example

Show that: 

Solution

This example uses the Maxwell Relations Table from the previous blog entry. You can read it here

Lets combine the first and second laws: dU = TdS - pdV

Divide both sides by dV and keep T as a constant:

Note that:

This results in:

Using the Maxwell Relation for A:

This results in:

Since:

This confirms that

Maxwell Relations

Background

Maxwell’s equations are important for electric and magnetic fields, while Maxwell’s relations focus on thermodynamic quantities like temperature, pressure, volume, and entropy and are important when analyzing thermodynamic systems like internal combustion engines.

Like Maxwell’s equations, Maxwell’s relations are based on partial derivatives. The relations describe the symmetry of second-order partial derivatives of thermodynamic quantities. Maxwell’s relations directly and indirectly cover eight thermodynamic quantities including:

• pressure (P),
• volume (V),
• temperature (T),
• entropy (S) measured in Joules per Kelvin (J/K),
• internal energy (U) measured in Joules,
• enthalpy (H),
• Helmholtz free energy (F or A),
• Gibbs free energy (G).

P, V, and T are simple quantities while the other thermodynamic quantities are more complex, thus Maxwell’s relations are very important.

For example, enthalpy (H) measures a system’s total heat content is defined as the sum of its internal energy (U) plus the product of pressure (P) and volume (V): H = U + PV.

Helmholtz free energy (F) measures the useful work obtainable from a closed system at constant temperature and volume and is defined as F = U – TS.

Gibbs free energy (G) is used to predict the spontaneity of a process at a given temperature and is defined as G = H – TS, or ∂G = ∂H – T∂S:

• When ∂G < 0, the process is spontaneous (favored).
• When ∂G > 0, the process is non-spontaneous (not favored).
• If ∂G = 0, the process is at equilibrium.

Mathematical Derivations of Maxwell's Relations

To derive Maxwell’s relations, it is useful to combine the First and Second Laws into a single mathematical statement. In order to do that, one notes that since:

for a reversible process: dq = TdS

Since: dw = TdS – pdV

for a reversible expansion in which only p-V works is done, and since dU = dq + dw

dU = TdS – pdV         (1)

This differential for dU can be used to simplify the differentials for H, A (Helmholtz energy) and G. But it is even more useful due to the constraints it places on the variables T, S, p and V due to the mathematics of exact differentials. 

Equation 1 suggests that the natural variables of internal energy are S and V or simply U (S, V). Thus, the total differential (dU) in equation 1 can be expressed as follows:

By studying equations 1 and 2, we can see that:

Since dU is an exact differential, the Euler relation must hold that:

substituting equations 3 and 4 into the above expression:

This is an example of a Maxwell Relation.

A similar result can be derived based on the definition of H:

H = U + pV

Using chain rule and differentiating d(pV) results in:

dH = dU + pdV + Vdp

Making the substitution using the combined first and second laws (dU = TdS - pdV) for a reversible change involving on expansion (p-V) work:

dH = TdS – pdV + pdV + Vdp

dH = TdS + Vdp         (5)

As in the case of internal energy, this suggests that the natural variables of H are S and p. Thus, 

If we compare equation 5 and 6 we get:

We can now see that equation 3 and equation 7 are equal to T hence are equal to each other:

Furthermore, the Euler Relation must also hold true:

This is the Maxwell Relation of H. The same can be done for G and A.

Summary

Maxwell Relations are summarized below:

Fundamentals of Partial Differentiation of the Exact Differential Equation

Carrying out partial differentiation of the Exact Differential Equation of M and N with respect to z and y, respectively, results in:

dx is a perfect differential when the above equation is satisfied for any function x.

Similarly, if y = y(x, z) and z = z(x, y), then from these two relations, we have:

In terms of p, v and T, the following relation holds true:

Thermodynamic Relations

Some properties like temperature, pressure, volume, and mass can be measured directly while other properties like density and specific volume can be determined from these using some simple relations.

However, properties like internal energy, enthalpy, and entropy are not so easy to determine as they cannot be measured directly or related to easily measurable properties through some simple relations.

Thus, it is important to develop some fundamental relations between commonly encountered thermodynamic properties and express the properties that cannot be measured directly in terms of easily measurable properties.

Partial Derivatives and Associated Relations

Most basic thermodynamic relations involve differentials. Let us consider a function f that depends on a single variable x, i.e., f = f(x) as shown below:

The derivative of the function at a point defined as Slope expressed as:

Therefore, the derivative of a function f(x) with respect to x represents the rate of change of f with x.

Let us now consider a function that depends on two (or more) variables, such as z = z(x, y). This time the value of z depends on both x and y.

It is sometimes necessary to examine the dependence of z on only one of the variables by allowing one variable to change while holding the others constant and observing the change in the function.

The variation of z(x, y) with x when y is held constant is called the partial derivative of z with respect to x, and it is expressed as:

To obtain a relation for the total differential change in z(x, y) for simultaneous changes in x and y would be:

The above equation is the fundamental relation for the total differential of a dependent variable in terms of its partial derivatives with respect to the independent variables. This relation can easily be extended to include more independent variables.

An important relation for partial derivatives is used in calculus to test whether a differential dz is exact or inexact. In thermodynamics, this relation forms the basis for the development of the Maxwell relations which was discussed in a previous blog entry here where dz is replaced by dφ.

We can now develop two important relations for partial derivatives – the reciprocity and the cyclic relations.

The function z = z(x, y) can also be expressed as x = x(y, z) if y and z are taken to be the independent variables. Then the total differential of x becomes:

Eliminating dx from the above equation by combining the dz and dx equations we get:

The variables y and z are independent of each other and thus can be varied independently. For example, y can be held constant (dy = 0), and z can be varied over a range of values (dz ≠ 0). Therefore, for this equation to be valid at all times, the terms in the brackets must equal zero, regardless of the values of y and z. 

Setting the terms in each bracket of the above equation equal to zero gives two equations:

This equation is called the reciprocity relation, and it shows that the inverse of a partial derivative is equal to its reciprocal.

This equation is called the cyclic relation, and it is frequently used in thermodynamics.

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: