Heat from the lens of thermodynamics

Let me start with specific heat. Specific heat is the heat that 1 gram substance requires to raise the temperature by 1 degc. The specific heat of a liquid or a solid is straightforward. However, it is complex when it comes to gas because a gas is a compressible fluid and involves both an increase in internal energy and work energy when heat is applied.? The specific heat of gas is expressed as Cp at constant pressure and Cv at constant volume. Both carry different meanings.

?I will focus on the specific heat of gases in this note.

There are four important equations for specific heat

Cp = dH/dT [ total energy change] [1st important equation]

Cv = dU/dT [ internal energy change] [2nd important equation]

Cp-Cv = nR [3rd important equation]

Cp/Cv = Y or dH/dU = Y [4th important equation]

Details

Step by step

Cp

Fundamentally when you heat a gas You increase the total energy of the gas at constant pressure called ‘enthalpy’

Cp = dH/dT [ total energy change] [1st important equation]

Cv

When you heat a gas at constant volume you increase the internal energy.

Cv = dU/dT [ internal energy change] [2nd important equation]

Therefore, since Cp of a gas is dH/dT

Cp = dH/dT

We can write this equation Cp dT = dH = dU + PdV + VdP

At constant pressure, Cp dT = dU + PdV

From the ideal gas equation PV = nRT,

At constant pressure PdV = n RdT

Substitute this in the above equation

CpdT = dU + nRdT ?[dU/dT = Cv]

Divide both sides by dT

Cp = Cv + n R

Cp-Cv = nR [3rd important equation]

This is one fundamental equation that gives the work the gas does or it separates work energy from the internal energy.

Cp/Cv = Y or dH/dU = Y [4th important equation]

Y is the polytropic quotient that comes from the equation PV^n = Constant

Cp/Cv equation is an important equation for adiabatic processes that signifies how much internal energy can extracted adiabatically from a gas.

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To understand heat transfer in a heat exchanger from the lens of thermodynamics, several fundamental principles come into play. Here are some of the key principles that govern the operation of heat exchangers: Conservation of Energy: This principle, also known as the First Law of Thermodynamics, states that energy cannot be created or destroyed but can only be transferred or converted from one form to another. In the context of a heat exchanger, energy is transferred between the hot and cold fluids without any net energy gain or loss.

Conservation of Mass: The Second Law of Thermodynamics dictates the conservation of mass. This principle ensures that the total mass entering the heat exchanger must be equal to the total mass exiting the heat exchanger. It emphasizes that mass cannot be created or destroyed but can only be transferred between the fluids.

Heat Transfer: Heat transfer principles play a pivotal role in heat exchangers. They establish the mechanism by which heat is transferred from one fluid to another. The most common modes of heat transfer are conduction, convection, and radiation. In a heat exchanger, heat is primarily transferred through conduction and convection between the hot and cold fluids.

Heat Exchange Rate: The rate at which heat is transferred between the two fluids is determined by the temperature difference between them. According to Fourier's Law of Heat Conduction, the rate of heat transfer is directly proportional to the temperature gradient. Therefore, a larger temperature difference between the hot and cold fluids will result in a higher heat exchange rate.

Entropy: Entropy is a measure of the degree of disorder or randomness in a system. The Second Law of Thermodynamics states that the entropy of an isolated system can only increase or remain constant. In the context of a heat exchanger, the Second Law implies that the heat transfer process results in an increase in the entropy of the system due to irreversibilities and losses.

Pressure Drop: A heat exchanger often causes a pressure drop in the fluid due to flow resistance. Pressure drop is a crucial consideration as excessive pressure drop can impact system performance and energy consumption. Thermodynamics plays a role in understanding and minimizing pressure drop by considering fluid properties, flow rates, and heat exchanger design.

Here are some typical equations that relate heat and thermodynamics: The

First Law of Thermodynamics: ?? ΔU = Q - W ?? This equation expresses the change in internal energy (ΔU) of a system as the difference between the heat added or removed (Q) and the work done on or by the system (W). ?Ideal Gas Law: ?? PV = nRT ?? The ideal gas law relates the pressure (P), volume (V), number of moles (n), gas constant (R), and temperature (T) of an ideal gas.

Carnot Efficiency Equation: ?? η_carnot = 1 - (Tc/Th) ?? This equation represents the maximum efficiency (η_carnot) of a heat engine operating between hot and cold reservoirs at temperatures Th and Tc, respectively.

Fourier's Law of Heat Conduction: ?? Q = -kA(dT/dx) ?? This equation describes the rate of heat transfer (Q) through a material with thermal conductivity (k), cross-sectional area (A), and temperature gradient (dT/dx).

Clausius-Clapeyron Equation: ?? ln(P2/P1) = -(ΔHvap/R)(1/T2 - 1/T1) ?? This equation relates the vapor pressure of a substance at two different temperatures (P1 and P2) using the enthalpy of vaporization (ΔHvap) and the gas constant (R). Nusselt Number: ?? Nu = (hL)/k ?? The Nusselt number (Nu) relates the convective heat transfer coefficient (h), characteristic length (L), and thermal conductivity (k) in convection heat transfer.

Reynolds Number: ?? Re = (ρVD)/μ ?? The Reynolds number (Re) characterizes the flow of a fluid with density (ρ), velocity (V), characteristic dimension (D), and dynamic viscosity (μ). It determines whether the flow is laminar or turbulent.