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Chapter 12

Conservation of Energy: The Energy Equation and the Thermodynamic Equation of State

12.1 Introduction

After discussing the conservation of momentum, we now turn to the last equation we need in order to calculate all six flow variables. This last equation is derived from the conservation of energy. For this equation, we need to sum up all energy contributions into and out of the control volume dV . Energy can be brought into and removed from the control volume

• in the form of energy flow by convection, i.e., in- and outflowing mass: Ė_convection

• in the form of heat flow by conduction, i.e., heat transfer across the boundary: ${\dot{Q}}_{conduction}$

• in the form of work performed by boundary forces, i.e., shear, normal forces as well as pressure acting on the surfaces of the control volume: Ẇ_{boundary forces}

• in the form of volume energy flow created by effects acting on the whole volume, e.g., chemical reactions or microwave heating: ${\dot{Q}}_{volume effect}$

• in the form of volume work flow created by volume forces, e.g., electrical or gravimetric forces: Ẇ_{volume forces}

The overall energy of the control volume will change over time as a consequence of these contribution. According to the first law of thermodynamics (see Eq. 6.32) the total energy of the system can be written as

$e_{V} = \frac{m_{i}}{V} (u_{V} + e_{V, pot} + e_{V, kin}) = (u_{V} + e_{V, pot} + e_{V, kin})$

(Eq. 12.1)

Please note that contrary to the convention used in section 6.6, the specific energy does not referred to the mass but to the volume (thus the index V). This makes sense because the density (and therefore the mass) of the control volume changes but the volume remains unchanged. However, comparing Eq. 12.1 with the first law of thermodynamics (see section 6.6), we can see that this is effectively the change of total energy of the system. e_{V, pot} is the specific potential energy resulting from gravimetric height changes, whereas e_{V, kin} is the specific kinetic energy resulting from changes in momentum.

We will restrict our discussion to control volumes in Eulerian frames of reference, i.e., to control volumes that are fixed in space. We therefore have no change in the potential energy, and Eq. 12.1 can therefore be simplified to

$e_{V} = ρ (u_{V} + e_{V, kin})$

(Eq. 12.2)

The internal energy can be expressed by the velocity and is given by

$e_{V kin} = \frac{1}{2} ρ {‖ v ‖}^{2} = \frac{1}{2} ρ (v_{x}^{2} + v_{y}^{2} + v_{x}^{2})$

We know that

E = e_V V

from which we follow

dE = e_V dV

Therefore the change of energy in the control volume over time can be expressed as

$\begin{array}{l} \frac{d E}{d t} & = & \frac{d e_{V}}{d t} d V = \frac{d e_{V}}{d t} d x d y d z \\ = & \frac{d}{d t} (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2})) d x d y d z \end{array}$

si6_e (Eq. 12.3)

In the following, we will look at each contribution to the change of energy of the control volume.

12.2 Energy Transfer by Convection

We will start by discussing the change of energy due to convection. Fig. 12.1 displays the mass influx and outflux into and out of the control volume. Each mass flow will bring along energy in the form of kinetic and internal energy. The energy flows are calculated as

f12-01-9781455731411 — Fig. 12.1 Energy change in the control volume as a consequence of mass in- and outflux by convection.

$d Ė_{convection} = e_{V} d \dot{V} = e_{V} v d A$

Along the individual axes, the energy flows are calculated to be in x-direction

$\begin{array}{l} d Ė_{x} & = & Ė_{x} - (Ė_{x} + \frac{\partial Ė_{x}}{\partial x} d x) \\ = & (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{x} - (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{x} + \frac{\partial (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{x})}{\partial x} d x)) d y d z \\ = & - \frac{\partial (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{x})}{\partial x} d x d y d z \end{array}$

si8_e

in y-direction

$\begin{array}{l} d Ė_{y} & = & Ė_{y} - (Ė_{y} + \frac{\partial Ė_{y}}{\partial y} d y) \\ = & (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{y} - (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{y} + \frac{\partial (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{y})}{\partial y} d y)) d x d z \\ = & - \frac{\partial (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{y})}{\partial y} d x d y d z \end{array}$

si9_e

in z-direction

$\begin{array}{l} d Ė_{z} & = & Ė_{z} - (Ė_{z} + \frac{\partial Ė_{z}}{\partial z} d z) \\ = & (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{z} - (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{z} + \frac{\partial (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{z})}{\partial z} d z)) d x d y \\ = & - \frac{\partial (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{z})}{\partial z} d x d y d z \end{array}$

si10_e

in summary

$d Ė_{conv.} = - (\frac{\partial (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{x})}{\partial x} + \frac{\partial (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{y})}{\partial y} + \frac{\partial (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{z})}{\partial z}) d x d y d z$

si11_e (Eq. 12.4)

As stated, the energy equation will amount to only one equation. So although we have made the balance along the individual axes, the overall energy contribution will be summed up in Eq. 12.4.

12.3 Heat Flow by Conduction

The second form of energy influx and outflux is due to heat flow in form of conduction across the surfaces of the control volume (see Fig. 12.2). These heat flows are calculated according to Fourier’s law of heat conduction (see Eq. 6.73) as

f12-02-9781455731411 — Fig. 12.2 Energy change in the control volume as a consequence of heat conduction across the surfaces.

$d {\dot{Q}}_{conduction} = - λ \frac{\partial T}{\partial n} d A$

with n being the normal vector of the area segment dA. Along the individual axes, the heat flows are calculated to be

in x-direction

$\begin{array}{l} d {\dot{Q}}_{x} & = & {\dot{Q}}_{x} - ({\dot{Q}}_{x} + \frac{\partial {\dot{Q}}_{x}}{\partial x} d x) = (- λ \frac{\partial T}{\partial x} - (- λ \frac{\partial T}{\partial x} + \frac{\partial}{\partial x} (λ \frac{\partial T}{\partial x}) d x)) d y d z \\ = & \frac{\partial}{\partial x} (λ \frac{\partial T}{\partial x}) d x d y d z \end{array}$

si13_e

in y-direction

$\begin{array}{l} d {\dot{Q}}_{y} & = & {\dot{Q}}_{y} - ({\dot{Q}}_{y} + \frac{\partial {\dot{Q}}_{y}}{\partial y} d y) = (- λ \frac{\partial T}{\partial y} - (- λ \frac{\partial T}{\partial y} + \frac{\partial}{\partial y} (λ \frac{\partial T}{\partial y}) d y)) d x d z \\ = & \frac{\partial}{\partial y} (λ \frac{\partial T}{\partial y}) d x d y d z \end{array}$

si14_e

in z-direction

$\begin{array}{l} d {\dot{Q}}_{z} & = & {\dot{Q}}_{z} - ({\dot{Q}}_{z} + \frac{\partial {\dot{Q}}_{z}}{\partial z} d z) = (- λ \frac{\partial T}{\partial z} - (- λ \frac{\partial T}{\partial z} + \frac{\partial}{\partial z} (λ \frac{\partial T}{\partial z}) d z)) d x d y \\ = & \frac{\partial}{\partial z} (λ \frac{\partial T}{\partial z}) d x d y d z \end{array}$

si15_e

in summary

$d {\dot{Q}}_{conduction} = - (\frac{\partial}{\partial x} (λ \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} (λ \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (λ \frac{\partial T}{\partial z})) d x d y d z$

si16_e (Eq. 12.5)

Again, the terms along the axes are summed up so that the total contribution of conduction is represented by Eq. 12.5.

12.4 Work Flow by Boundary Forces

The third term is the work performed by shear and normal forces that act on the surfaces of the control volume (see Fig. 12.3). Again, we will split the normal forces into two contributions: one contribution resulting from friction forces of the liquid σ_ii and a contribution resulting from the action of pressure p. We introduced this concept before (see Eq. 11.13, Eq. 11.14, and Eq. 11.15).

f12-03-9781455731411 — Fig. 12.3 Contribution to the change of energy by the action of surface and normal forces as well as pressure on the boundary of the control volume.

The work due to the action of the forces is calculated according to Eq. 6.33 as dẆ_{boundary forces} = {σ, τ, p} dV = {σ, τ, p} v dA

Along the individual axes the work is calculated to be on surface dy dz

$\begin{array}{l} d Ẇ_{x} & = & Ẇ_{x} - (Ẇ_{x} + \frac{\partial Ẇ_{x}}{\partial x} d x) + [p v - (p v + \frac{\partial (p v)}{\partial x} d x) - σ_{x x} v + (σ_{x x} v + \frac{\partial (σ_{x x} v)}{\partial x} d x) - τ_{x y} v + (τ_{x y} v + \frac{\partial (τ_{x y} v)}{\partial x} d x) - τ_{x z} v + (τ_{x z} v + \frac{\partial (τ_{x z} v)}{\partial x} d x)] d y d z \\ = & (- \frac{\partial (p v_{x})}{\partial x} + \frac{\partial (σ_{x x} v)}{\partial x} + \frac{\partial (τ_{x y} v)}{\partial x} + \frac{\partial (τ_{x z} v)}{\partial x}) d x d y d z \end{array}$

si17_e

on surface dx dz

$\begin{array}{l} d Ẇ_{y} & = & Ẇ_{y} - (Ẇ_{y} + \frac{\partial Ẇ_{y}}{\partial y} d y) + [p v - (p v + \frac{\partial (p v)}{\partial y} d y) - σ_{y y} v + (σ_{y y} v + \frac{\partial (σ_{y y} v)}{\partial y} d y) - τ_{y x} v + (τ_{y x} v + \frac{\partial (τ_{y x} v)}{\partial y} d y) - τ_{y z} v + (τ_{y z} v + \frac{\partial (τ_{y z} v)}{\partial y} d y)] d x d z \\ = & (- \frac{\partial (p v_{y})}{\partial y} + \frac{\partial (σ_{y y} v)}{\partial y} + \frac{\partial (τ_{y x} v)}{\partial y} + \frac{\partial (τ_{y z} v)}{\partial y}) d x d y d z \end{array}$

si18_e

on surface dx dy

$\begin{array}{l} d W_{z} & = & W_{z} - (W_{z} + \frac{\partial W_{z}}{\partial z} d z) + [p v - (p v + \frac{\partial (p v)}{\partial z} d z) - σ_{z z} v + (σ_{z z} v + \frac{\partial (σ_{z z} v)}{\partial z} d z) - τ_{z x} v + (τ_{z x} v + \frac{\partial (τ_{z x} v)}{\partial z} d z) - τ_{z y} v + (τ_{z y} v + \frac{\partial (τ_{z y} v)}{\partial z} d z)] d x d y \\ = & (- \frac{\partial (p v_{z})}{\partial z} + \frac{\partial (σ_{z z} v)}{\partial z} + \frac{\partial (τ_{z x} v)}{\partial z} + \frac{\partial (τ_{z y} v)}{\partial z}) d x d y d z \end{array}$

si19_e

in summary

$d Ẇ_{boundary forces} = [- \frac{\partial (p v_{x})}{\partial x} + \frac{\partial (σ_{x x} v)}{\partial x} + \frac{\partial (τ_{x y} v)}{\partial x} + \frac{\partial (τ_{x z} v)}{\partial x} - \frac{\partial (p v_{y})}{\partial y} + \frac{\partial (σ_{y y} v)}{\partial y} + \frac{\partial (τ_{y x} v)}{\partial y} + \frac{\partial (τ_{y z} v)}{\partial y} - \frac{\partial (p v_{z})}{\partial z} + \frac{\partial (σ_{z z} v)}{\partial z} + \frac{\partial (τ_{z x} v)}{\partial z} + \frac{\partial (τ_{z y} v)}{\partial z}] d x d y d z$

si20_e (Eq. 12.6)

Again, the individual terms can be summed up to Eq. 12.6.

12.5 Heat Flow by Volume Effects

The fourth contribution we need to consider is the heat flow by volume effects. Typical examples include chemical reactions happening inside the control volume. In contrast to the contributions of conduction, these effects act on the whole volume. Therefore we do not need to consider the changes of the heat flow as we move along the individual axes, and therefore we do not have to derive the partial differentials. The unit of q_V is J m⁻³. Analytically this contribution is described by

$\begin{array}{l} d {\dot{Q}}_{volume effect} & = & \dot{q} d m = \dot{q} ρ d V \\ = & \frac{\dot{Q}}{m} ρ d V \\ = & \frac{\dot{Q}}{V} d V \\ = & \dot{q} V d V \\ = & \dot{q} V d x d y d z \end{array}$

si21_e (Eq. 12.7)

where $\dot{q} V$ is the specific heat flow per volume in unit J m⁻³ s⁻¹.

12.6 Work Flow by Volume Forces

The fifth and last contribution is the work flow created by the action of volume forces $\vec{k}$ , e.g., electrical forces that act on all ions in the volume as well as gravimetric forces that act on the mass in the control volume. In comparison to the heat flow because of volume effects (see section 12.5), we do not have to consider the changes along the individual axes because volume effects are not considered to change. The unit of $\vec{k}$ is N m⁻³. The contribution along the individual axes are calculated as

in x-direction

dẆ_{volume forces, x} = k_x v_x dx dy dz

in y-direction

dẆ_{volume forces, y} = k_y v_y dx dy dz

in z-direction

dẆ_{volume forces, z} = k_z v_z dx dy dz

in summary

$d Ẇ_{volume forces} = (k_{x} v_{x} + k_{y} v_{y} + k_{z} v_{z}) d x d y d z = (\vec{k} \cdot \vec{v}) d x d y d z$

si25_e (Eq. 12.8)

12.7 Balance of Contributions

Summing up Eq. 12.4, Eq. 12.5, Eq. 12.6, Eq. 12.7, and Eq. 12.8 and substituting them in Eq. 12.3, we obtain the following expression for the conservation of energy after dividing by dx dy dz:

$\begin{array}{l} \frac{d}{d t} (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2})) \\ = - \frac{\partial (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{x})}{\partial x} + \frac{\partial (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{y})}{\partial y} + \frac{\partial (ρ (u_{V} + \frac{1}{2} ‖ v ‖^{2}) v_{z})}{\partial z} & convection \\ + \frac{\partial}{\partial x} (λ \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} + (λ \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (λ \frac{\partial T}{\partial z}) & conduction \\ - \frac{\partial (p v_{x})}{\partial x} + \frac{\partial (σ_{x x} v)}{\partial x} + \frac{\partial (τ_{x y} v)}{\partial x} + \frac{\partial (τ_{x z} v)}{\partial x} \\ - \frac{\partial (p v_{y})}{\partial y} + \frac{\partial (σ_{y y} v)}{\partial y} + \frac{\partial (τ_{y x} v)}{\partial y} + \frac{\partial (τ_{y z} v)}{\partial y} \\ - \frac{\partial (p v_{z})}{\partial z} + \frac{\partial (σ_{z z} v)}{\partial z} + \frac{\partial (τ_{z x} v)}{\partial z} + \frac{\partial (τ_{z y} v)}{\partial z} & surface forces \\ + ρ {\dot{q}}_{V} & vol. heat flow \\ + \vec{k \cdot} \vec{ v} & vol. work flow \end{array}$

si26_e (Eq. 12.9)

12.7.1 Simplifications on the Left-Hand Side

Eq. 12.9 describes all the physics of the conservation of energy. However, the equation can still be simplified. We first combine the left-hand side of the equation with the first term of the right-hand side.

$\frac{d}{d t} (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2})) + \frac{\partial (ρ (u v + \frac{1}{2} {‖ v ‖}^{2}) v_{x})}{\partial x} + \frac{\partial (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{y})}{\partial y} + \frac{\partial (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{z})}{\partial z}$

si27_e (Eq. 12.10)

Now we expand the partial differentials of the left-hand side of this equation according to Eq. 3.13.

$\begin{array}{l} \frac{d}{d t} (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2})) + \frac{\partial (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{x})}{\partial x} + \frac{\partial (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{y})}{\partial y} + \frac{\partial (ρ (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) v_{z})}{\partial z} \\ = (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) \frac{\partial ρ}{\partial t} + ρ \frac{\partial (u_{V} + \frac{1}{2} {‖ v ‖}^{2})}{\partial t} + (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) (\frac{\partial (ρ v_{x})}{\partial x} + \frac{\partial (ρ v_{y})}{\partial y} + \frac{\partial (ρ v_{z})}{\partial z}) + \\ (ρ v_{x}) \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial x} + (ρ v_{y}) \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial y} + (ρ v_{z}) \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial z} \\ = & (u_{V} + \frac{1}{2} {‖ v ‖}^{2}) (\frac{\partial ρ}{\partial t} + \frac{\partial (ρ v_{x})}{\partial x} + \frac{\partial (ρ v_{y})}{\partial y} + \frac{\partial (ρ v_{z})}{\partial z}) + \\ ρ (\frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial t} + v_{x} \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial x} + v_{y} \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial y} + v_{z} \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial z}) \\ = & ρ (\frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial t} + v_{x} \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial x} + v_{y} \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial y} + v_{z} \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial z}) \end{array}$

si28_e (Eq. 12.11)

The last simplification can be made because

$\frac{\partial ρ}{\partial t} + \frac{\partial (ρ v_{x})}{\partial x} + \frac{\partial (ρ v_{y})}{\partial y} + \frac{\partial (ρ v_{z})}{\partial z} = 0$

(Eq. 12.12)

due to the continuity equation (see Eq. 10.16). We can expand (see Eq. 12.11) a little further by using

$\begin{array}{l} \frac{\partial}{\partial i} (\frac{1}{2} {‖ v ‖}^{2}) & = \frac{1}{2} \frac{\partial (v_{x}^{2} + v_{y}^{2} + v_{z}^{2})}{\partial i} = \frac{1}{2} (\frac{\partial v_{x}^{2}}{\partial x} + \frac{\partial v_{y}^{2}}{\partial y} + \frac{\partial v_{z}^{2}}{\partial z}) \\ = \frac{1}{2} (2 v_{x} \frac{\partial v_{x}}{\partial i} + 2 v_{y} \frac{\partial v_{y}}{\partial i} + 2 v_{z} \frac{\partial v_{z}}{\partial i}) \\ = v_{x} \frac{\partial v_{x}}{\partial i} + v_{y} \frac{\partial v_{y}}{\partial i} + v_{z} \frac{\partial v_{z}}{\partial i} \end{array}$

si30_e

with i = {t, x, y, z}. Therefore

$\begin{array}{l} \frac{\partial}{\partial t} (\frac{1}{2} {‖ v ‖}^{2}) & = & v_{x} \frac{\partial v_{x}}{\partial t} + v_{y} \frac{\partial v_{y}}{\partial t} + v_{z} \frac{\partial v_{z}}{\partial t} \\ \frac{\partial}{\partial x} (\frac{1}{2} {‖ v ‖}^{2}) & = & v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial x} + v_{z} \frac{\partial v_{z}}{\partial x} \\ \frac{\partial}{\partial y} (\frac{1}{2} {‖ v ‖}^{2}) & = & v_{x} \frac{\partial v_{x}}{\partial y} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial y} \\ \frac{\partial}{\partial z} (\frac{1}{2} {‖ v ‖}^{2}) & = & v_{x} \frac{\partial v_{x}}{\partial z} + v_{y} \frac{\partial v_{y}}{\partial z} + v_{z} \frac{\partial v_{z}}{\partial z} \end{array}$

si31_e

We can use these simplifications during expansion of the partial differentials in Eq. 12.11 to find

$\begin{array}{l} ρ (\frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial t} + v_{x} \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial x} + v_{y} \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial y} + v_{z} \frac{\partial u_{V} + \frac{1}{2} {‖ v ‖}^{2}}{\partial z}) \\ = ρ (\frac{\partial u_{V}}{\partial t} + \frac{\partial \frac{1}{2} {‖ v ‖}^{2}}{\partial t} + v_{x} (\frac{\partial u_{V}}{\partial x} + \frac{\partial \frac{1}{2} {‖ v ‖}^{2}}{\partial x}) + v_{y} (\frac{\partial u_{V}}{\partial y} + \frac{\partial \frac{1}{2} {‖ v ‖}^{2}}{\partial y}) + v_{z} (\frac{\partial u_{V}}{\partial z} + \frac{\partial \frac{1}{2} {‖ v ‖}^{2}}{\partial z})) \\ = ρ (\frac{\partial u_{V}}{\partial t} + v_{x} \frac{\partial v_{x}}{\partial t} + v_{y} \frac{\partial v_{y}}{\partial t} + v_{z} \frac{\partial v_{z}}{\partial t} \\ + v_{x} (\frac{\partial u_{V}}{\partial x} + v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial x} + v_{z} \frac{\partial v_{z}}{\partial x}) \\ + v_{y} (\frac{\partial u_{V}}{\partial y} + v_{x} \frac{\partial v_{x}}{\partial y} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial y}) \\ + v_{z} (\frac{\partial u_{V}}{\partial z} + v_{x} \frac{\partial v_{x}}{\partial z} + v_{y} \frac{\partial v_{y}}{\partial z} + v_{z} \frac{\partial v_{z}}{\partial z})) \end{array}$

si32_e (Eq. 12.13)

12.7.2 Simplifications on the Right-Hand Side

This is as far as we can go with simplifying the left-hand side. We now turn to the right-hand side of Eq. 12.9. The conduction terms as well as the contribution from volume heat flow will remain essentially unchanged but we can apply a couple of simplifications on the remaining terms. We will start with the terms originating from the boundary forces and the work performed by volume forces.

$\begin{array}{l} - \frac{\partial (p v_{x})}{\partial x} + \frac{\partial (σ_{x x} v)}{\partial x} + \frac{\partial (τ_{x y} v)}{\partial x} + \frac{\partial (τ_{x z} v)}{\partial x} \\ - \frac{\partial (p v_{y})}{\partial y} + \frac{\partial (σ_{y y} v)}{\partial y} + \frac{\partial (τ_{y x} v)}{\partial y} + \frac{\partial (τ_{y z} v)}{\partial y} \\ - \frac{\partial (p v_{z})}{\partial z} + \frac{\partial (σ_{z z} v)}{\partial z} + \frac{\partial (τ_{z x} v)}{\partial z} + \frac{\partial (τ_{z y} v)}{\partial z} \\ = - (p (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}) + v_{x} \frac{\partial p}{\partial x} + v_{y} \frac{\partial p}{\partial y} + v_{z} \frac{\partial p}{\partial z}) \\ + v_{x} (\frac{\partial σ_{x x}}{\partial x} + \frac{\partial σ_{y x}}{\partial y} + \frac{\partial σ_{z x}}{\partial x}) + σ_{x x} \frac{\partial v_{x}}{\partial x} + τ_{y x} \frac{\partial v_{x}}{\partial y} + σ_{z x} \frac{\partial v_{x}}{\partial z} \\ + v_{y} (\frac{\partial σ_{y y}}{\partial y} + \frac{\partial σ_{x y}}{\partial x} + \frac{\partial σ_{z y}}{\partial y}) + σ_{y y} \frac{\partial v_{y}}{\partial y} + τ_{x y} \frac{\partial v_{y}}{\partial x} + σ_{z y} \frac{\partial v_{y}}{\partial z} \\ + v_{z} (\frac{\partial σ_{z z}}{\partial z} + \frac{\partial σ_{x z}}{\partial x} + \frac{\partial σ_{y z}}{\partial z}) + σ_{z z} \frac{\partial v_{z}}{\partial z} + τ_{x z} \frac{\partial v_{z}}{\partial x} + σ_{y x} \frac{\partial v_{z}}{\partial y} \\ + k_{x} v_{x} + k_{y} v_{y} + k_{z} v_{z} \end{array}$

si33_e (Eq. 12.14)

Now we will use a couple of equations that we have used before. Eq. 11.19, Eq. 11.20, and Eq. 11.21 state that we can make the following simplifications:

$\begin{array}{l} \frac{\partial ρ v_{x}}{\partial t} + \frac{\partial (ρ v_{x} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial y} + \frac{\partial (ρ v_{z} v_{x})}{\partial z} = \frac{\partial σ_{x x}}{\partial x} - \frac{\partial p}{\partial x} + \frac{\partial τ_{y x}}{\partial y} + \frac{\partial τ_{z x}}{\partial z} + k_{x} \\ \frac{\partial ρ v_{y}}{\partial t} + \frac{\partial (ρ v_{x} v_{y})}{\partial x} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z} = \frac{\partial σ_{y y}}{\partial y} - \frac{\partial p}{\partial y} + \frac{\partial τ_{x y}}{\partial x} + \frac{\partial τ_{z y}}{\partial z} + k_{y} \\ \frac{\partial ρ v_{z}}{\partial t} + \frac{\partial (ρ v_{x} v_{z})}{\partial x} + \frac{\partial (ρ v_{y} v_{z})}{\partial y} + \frac{\partial (ρ v_{z} v_{z})}{\partial z} = \frac{\partial σ_{z z}}{\partial z} - \frac{\partial p}{\partial z} + \frac{\partial τ_{x z}}{\partial x} + \frac{\partial τ_{y z}}{\partial y} + k_{z} \end{array}$

si34_e

If we rewrite Eq. 12.11, we can see that we can effectively make these simplifications:

$\begin{array}{l} - p (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}) \\ + v_{x} (- \frac{\partial p}{\partial x} + \frac{\partial σ_{x x}}{\partial x} + \frac{\partial τ_{y x}}{\partial y} + \frac{\partial τ_{z x}}{\partial z} + k_{x}) + σ_{x x} \frac{\partial v_{x}}{\partial x} + τ_{y x} \frac{\partial v_{x}}{\partial y} + τ_{z x} \frac{\partial v_{x}}{\partial z} \\ + v_{x} (- \frac{\partial p}{\partial y} + \frac{\partial σ_{y y}}{\partial y} + \frac{\partial τ_{x y}}{\partial x} + \frac{\partial τ_{z y}}{\partial z} + k_{y}) + σ_{y y} \frac{\partial v_{y}}{\partial y} + τ_{x y} \frac{\partial v_{y}}{\partial x} + τ_{z y} \frac{\partial v_{y}}{\partial z} \\ + v_{x} (- \frac{\partial p}{\partial z} + \frac{\partial σ_{z z}}{\partial z} + \frac{\partial τ_{x z}}{\partial x} + \frac{\partial τ_{y z}}{\partial y} + k_{z}) + σ_{z z} \frac{\partial v_{z}}{\partial z} + τ_{x z} \frac{\partial v_{z}}{\partial x} + τ_{y z} \frac{\partial v_{z}}{\partial y} \\ = - p (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}) \\ + v_{x} (\frac{\partial ρ v_{x}}{\partial t} + \frac{\partial (ρ v_{x} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial y} + \frac{\partial (ρ v_{z} v_{x})}{\partial z}) + σ_{x x} \frac{\partial v_{x}}{\partial x} + τ_{y x} \frac{\partial v_{x}}{\partial y} + τ_{z x} \frac{\partial v_{x}}{\partial z} \\ + v_{y} (\frac{\partial ρ v_{y}}{\partial t} + \frac{\partial (ρ v_{x} v_{y})}{\partial x} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z}) + σ_{y y} \frac{\partial v_{y}}{\partial y} + τ_{x y} \frac{\partial v_{y}}{\partial x} + τ_{z y} \frac{\partial v_{y}}{\partial z} \\ + v_{z} (\frac{\partial ρ v_{z}}{\partial t} + \frac{\partial (ρ v_{x} v_{z})}{\partial x} + \frac{\partial (ρ v_{y} v_{z})}{\partial y} + \frac{\partial (ρ v_{z} v_{z})}{\partial z}) + σ_{z z} \frac{\partial v_{z}}{\partial z} + τ_{x z} \frac{\partial v_{z}}{\partial x} + τ_{y z} \frac{\partial v_{z}}{\partial z} \end{array}$

si35_e (Eq. 12.15)

We will now expand the partial differentials in the brackets whereby we can again use the simplification given by the continuity equation (see Eq. 12.12).

$\begin{array}{l} \frac{\partial ρ v_{x}}{\partial t} + \frac{\partial (ρ v_{x} v_{x})}{\partial x} + \frac{\partial (ρ v_{y} v_{x})}{\partial y} + \frac{\partial (ρ v_{z} v_{x})}{\partial z} \\ = v_{x} (\frac{\partial ρ}{\partial t} + \frac{\partial^{2} (ρ v_{x})}{\partial x \partial x} + \frac{\partial^{2} (ρ v_{y})}{\partial y \partial y} + \frac{\partial^{2} (ρ v_{z})}{\partial z \partial z}) + ρ \frac{\partial v_{x}}{\partial t} + (ρ v_{x}) \frac{\partial v_{x}}{\partial x} + (ρ v_{y}) \frac{\partial v_{x}}{\partial y} + (ρ v_{z}) \frac{\partial v_{x}}{\partial z} \\ = ρ \frac{\partial v_{x}}{\partial t} + (ρ v_{x}) \frac{\partial v_{x}}{\partial x} + (ρ v_{y}) \frac{\partial v_{x}}{\partial y} + (ρ v_{z}) \frac{\partial v_{x}}{\partial z} \end{array}$

si36_e (Eq. 12.16)

$\begin{array}{l} \frac{\partial ρ v_{y}}{\partial t} + \frac{\partial (ρ v_{x} v_{y})}{\partial x} + \frac{\partial (ρ v_{y} v_{y})}{\partial y} + \frac{\partial (ρ v_{z} v_{y})}{\partial z} \\ = v_{y} (\frac{\partial ρ}{\partial t} + \frac{\partial^{2} (ρ v_{x})}{\partial x \partial x} + \frac{\partial^{2} (ρ v_{y})}{\partial y \partial y} + \frac{\partial^{2} (ρ v_{z})}{\partial z \partial z}) + ρ \frac{\partial v_{y}}{\partial t} + (ρ v_{x}) \frac{\partial v_{y}}{\partial x} + (ρ v_{y}) \frac{\partial v_{y}}{\partial y} + (ρ v_{z}) \frac{\partial v_{y}}{\partial z} \\ = ρ \frac{\partial v_{y}}{\partial t} + (ρ v_{x}) \frac{\partial v_{y}}{\partial x} + (ρ v_{y}) \frac{\partial v_{y}}{\partial y} + (ρ v_{z}) \frac{\partial v_{y}}{\partial z} \end{array}$

si37_e (Eq. 12.17)

$\begin{array}{l} \frac{\partial ρ v_{z}}{\partial t} + \frac{\partial (ρ v_{x} v_{z})}{\partial x} + \frac{\partial (ρ v_{y} v_{z})}{\partial y} + \frac{\partial (ρ v_{z} v_{z})}{\partial z} \\ = v_{z} (\frac{\partial ρ}{\partial t} + \frac{\partial^{2} (ρ v_{x})}{\partial x \partial x} + \frac{\partial^{2} (ρ v_{y})}{\partial y \partial y} + \frac{\partial^{2} (ρ v_{z})}{\partial z \partial z}) + ρ \frac{\partial v_{z}}{\partial t} + (ρ v_{x}) \frac{\partial v_{z}}{\partial x} + (ρ v_{y}) \frac{\partial v_{z}}{\partial y} + (ρ v_{z}) \frac{\partial v_{z}}{\partial z} \\ = ρ \frac{\partial v_{z}}{\partial t} + (ρ v_{x}) \frac{\partial v_{z}}{\partial x} + (ρ v_{y}) \frac{\partial v_{z}}{\partial y} + (ρ v_{z}) \frac{\partial v_{z}}{\partial z} \end{array}$

si38_e (Eq. 12.18)

So substituting Eq. 12.16, Eq. 12.17, and Eq. 12.18 into Eq. 12.15 results in the following term

$\begin{array}{l} - p (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{x}}{\partial x}) \\ + v_{x} (ρ \frac{\partial v_{x}}{\partial t} + (ρ v_{x}) \frac{\partial v_{x}}{\partial x} + (ρ v_{y}) \frac{\partial v_{x}}{\partial y} + (ρ v_{z}) + \frac{\partial v_{x}}{\partial z}) + σ_{x x} \frac{\partial v_{x}}{\partial x} + τ_{y x} \frac{\partial v_{x}}{\partial y} + τ_{z x} \frac{\partial v_{x}}{\partial z} \\ + v_{y} (ρ \frac{\partial v_{y}}{\partial t} + (ρ v_{x}) \frac{\partial v_{y}}{\partial x} + (ρ v_{y}) \frac{\partial v_{x}}{\partial y} + (ρ v_{z}) + \frac{\partial v_{_{y}}}{\partial z}) + σ_{y y} \frac{\partial v_{y}}{\partial_{y}} + τ_{x y} \frac{\partial v_{y}}{\partial x} + τ_{z y} \frac{\partial v_{y}}{\partial z} \\ + v_{z} (ρ \frac{\partial v_{z}}{\partial t} + (ρ v_{x}) \frac{\partial v_{z}}{\partial x} + (ρ v_{y}) \frac{\partial v_{z}}{\partial y} + (ρ v_{z}) + \frac{\partial v_{z}}{\partial z}) + σ_{z z} \frac{\partial v_{z}}{\partial_{z}} + τ_{x z} \frac{\partial v_{z}}{\partial x} + τ_{y z} \frac{\partial v_{z}}{\partial y} \end{array}$

si39_e

which we reintroduce in the right-hand side of Eq. 12.9. Setting this equation against the modified left-hand side given in Eq. 12.13 results in

$\begin{array}{l} ρ (\frac{\partial u_{V}}{\partial t} + v_{x} \frac{\partial v_{x}}{\partial t} + v_{y} \frac{\partial v_{y}}{\partial t} + v_{z} \frac{\partial v_{z}}{\partial t} + v_{x} (\frac{\partial u_{V}}{\partial x} + v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{\partial v_{y}}{\partial x} + v_{z} \frac{\partial v z}{\partial x}) \\ + v_{y} (\frac{\partial u_{V}}{\partial y} + v_{x} \frac{\partial v_{x}}{\partial y} + v_{y} \frac{\partial v_{y}}{\partial y} + v_{z} \frac{\partial v_{z}}{\partial y}) + v_{z} (\frac{\partial u_{V}}{\partial z} + v_{x} \frac{\partial v_{x}}{\partial z} + v_{y} \frac{\partial v_{y}}{\partial z} + v_{z} \frac{\partial v_{z}}{\partial z})) \\ = \frac{\partial}{\partial x} (λ \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} (λ \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (λ \frac{\partial T}{\partial z}) - p (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}) \\ + v_{x} (ρ \frac{\partial v_{x}}{\partial t} + (ρ v_{x}) \frac{\partial v_{x}}{\partial x} + (ρ v_{y}) \frac{\partial v_{x}}{\partial y} + (ρ v_{z}) \frac{\partial v_{x}}{\partial z}) + σ_{x x} \frac{\partial v_{x}}{\partial x} + τ_{y x} \frac{\partial v_{x}}{\partial y} + v_{z x} \frac{\partial v_{x}}{\partial z} \\ + v_{y} (ρ \frac{\partial v_{y}}{\partial t} + (ρ v_{x}) \frac{\partial v_{y}}{\partial x} + (ρ v_{y}) \frac{\partial v_{y}}{\partial y} + (ρ v_{z}) \frac{\partial v_{y}}{\partial z}) + σ_{y y} \frac{\partial v_{y}}{\partial y} + τ_{x y} \frac{\partial v_{y}}{\partial x} + τ_{z y} \frac{\partial v_{y}}{\partial z} \\ + v_{z} (ρ \frac{\partial v_{z}}{\partial t} + (ρ v_{x}) \frac{\partial v_{z}}{\partial x} + (ρ v_{y}) \frac{\partial v_{z}}{\partial y} + (ρ v_{z}) \frac{\partial v_{z}}{\partial z}) + σ_{z z} \frac{\partial v_{z}}{\partial z} + τ_{x z} \frac{\partial v_{z}}{\partial z} + τ_{y z} \frac{\partial v_{z}}{\partial y} \\ + {\dot{q}}_{v} \end{array}$

si40_e (Eq. 12.19)

As we can see, a number of terms can be dropped, after which the equation simplifies to

$\begin{array}{l} ρ (\frac{\partial u_{V}}{\partial t} + v_{x} \frac{\partial u_{V}}{\partial x} + v_{y} \frac{\partial u_{V}}{\partial y} + v_{z} \frac{\partial u_{V}}{\partial z}) \\ = \frac{\partial}{\partial x} (λ \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} (λ \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (λ \frac{\partial T}{\partial z}) - p (\dot{\nabla} \cdot \vec{v}) + ρ \dot{q} V + σ_{x x} \frac{\partial v_{x}}{\partial x} + τ_{y x} \frac{\partial v_{x}}{\partial y} + τ_{z x} \frac{\partial v_{x}}{\partial z} \\ + σ_{y y} \frac{\partial v_{y}}{\partial y} + τ_{x y} \frac{\partial v_{y}}{\partial x} + τ_{z y} \frac{\partial v_{y}}{\partial z} + σ_{z z} \frac{\partial v_{z}}{\partial z} + τ_{x z} \frac{\partial v_{z}}{\partial x} + τ_{y z} \frac{\partial v_{z}}{\partial y} \end{array}$

si41_e (Eq. 12.20)

where we used the vector notation

$p (\dot{\nabla} \cdot \vec{v}) = p (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z})$

si42_e

Eq. 12.20 is the conservation of energy in its most general form.

12.7.3 Dissipation Coefficient

The equation is often further structured by combining all normal and shear force terms into a dissipation coefficient Ψ. These terms combine the influence of fluid friction and thereby the loss of energy because of internal friction, a process referred to as dissipation. In order to do so, we will again convert the normal and shear forces into functions of the shear rate, i.e., the partial derivatives of the velocity terms using the Stokes’ friction approach given by Eq. 11.22, Eq. 11.23, Eq. 11.24, Eq. 11.25, Eq. 11.26, and Eq. 11.27. Introducing Ψ allows us to rewrite (see Eq. 12.20) as

si43_e (Eq. 12.21)

where Ψ is given as

$\begin{array}{l} η Ψ & = & σ_{x x} \frac{\partial v_{x}}{\partial x} + τ_{y x} \frac{\partial v_{x}}{\partial y} + τ_{z x} \frac{\partial v_{x}}{\partial z} + σ_{y y} \frac{\partial v_{y}}{\partial y} + τ_{x y} \frac{\partial v_{y}}{\partial x} + τ_{z y} \frac{\partial v_{y}}{\partial z} + σ_{z z} \frac{\partial v_{z}}{\partial z} + τ_{x z} \frac{\partial v_{z}}{\partial x} + τ_{y z} \frac{\partial v_{z}}{\partial y} \\ = & (2 η \frac{\partial v_{x}}{\partial x} - \frac{2}{3} η (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z})) \frac{\partial v_{x}}{\partial x} + (η (\frac{\partial v_{y}}{\partial x} + \frac{\partial v_{x}}{\partial y})) \frac{\partial v_{x}}{\partial y} + (η (\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x})) \frac{\partial v_{x}}{\partial z} \\ + (2 η \frac{\partial v_{y}}{\partial y} - \frac{2}{3} η (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z})) \frac{\partial v_{y}}{\partial y} + (η (\frac{\partial v_{y}}{\partial x} + \frac{\partial v_{x}}{\partial y})) \frac{\partial v_{y}}{\partial x} + (η (\frac{\partial v_{z}}{\partial y} + \frac{\partial v_{y}}{\partial z})) \frac{\partial v_{y}}{\partial z} \\ + (2 η \frac{\partial v_{z}}{\partial z} - \frac{2}{3} η (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z})) \frac{\partial v_{z}}{\partial z} + (η (\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x})) \frac{\partial v_{z}}{\partial x} + (η (\frac{\partial v_{z}}{\partial y} + \frac{\partial v_{y}}{\partial z})) \frac{\partial v_{z}}{\partial y} \\ = & η {(2 ({(\frac{\partial v_{x}}{\partial x})}^{2} + {(\frac{\partial v_{y}}{\partial y})}^{2} + {(\frac{\partial v_{z}}{\partial z})}^{2}) - \frac{2}{3} {(\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z})}^{2} + {(\frac{\partial v_{y}}{\partial x} + \frac{\partial v_{x}}{\partial y})}^{2} + {(\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x})}^{2} + (\frac{\partial v_{z}}{\partial y} + \frac{\partial v_{y}}{\partial z}))}^{2} \end{array}$

si44_e

which simplifies to

$Ψ = 2 ({(\frac{\partial v_{x}}{\partial x})}^{2} + {(\frac{\partial v_{y}}{\partial y})}^{2} + {(\frac{\partial v_{z}}{\partial z})}^{2}) - \frac{2}{3} {(\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z})}^{2} + {(\frac{\partial v_{y}}{\partial x} + \frac{\partial v_{x}}{\partial y})}^{2} + {(\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x})}^{2} + {(\frac{\partial v_{z}}{\partial y} + \frac{\partial v_{y}}{\partial z})}^{2}$

si45_e (Eq. 12.22)

12.8 Thermodynamic Equation of State

As stated in section 10.1.1 the last equation required in order to derive all field variables is a thermodynamic equation of state. This equation allows us to express the internal energy terms in Eq. 12.21 as functions of the other field variables, i.e., pressure p, density ρ, temperature T, and the velocity $\vec{v}$ . In the following example, we will employ the thermodynamic equation of state for an ideal gas that is given by Eq. 6.58. Using this equation, we can write the partial differentials of the internal energy U with respect to i = {T, x, y, z} as

$\begin{array}{l} \frac{\partial u}{\partial i} & = & c_{p} \frac{\partial T}{\partial i} - (\frac{1}{ρ} \frac{\partial p}{\partial i} + p \frac{\partial}{\partial i} (\frac{1}{ρ})) \\ = & c_{p} \frac{\partial T}{\partial i} - (\frac{1}{ρ} \frac{\partial p}{\partial i} - \frac{p}{ρ^{2}} \frac{\partial ρ}{\partial i}) \\ = & c_{p} \frac{\partial T}{\partial i} - \frac{1}{ρ} \frac{\partial p}{\partial i} + \frac{p}{ρ^{2}} \frac{\partial ρ}{\partial i} \end{array}$

si47_e

where we used the chain rule of differentiation (see Eq. 3.14) to expand the partial differential $\frac{\partial \frac{1}{ρ}}{\partial i}$ . Now we can rewrite Eq. 12.21 as

$\begin{array}{l} ρ (c_{p} \frac{\partial T}{\partial t} - \frac{1}{ρ} \frac{\partial p}{\partial t} + \frac{p}{ρ^{2}} \frac{\partial ρ}{\partial t} + v_{x} (c_{p} \frac{\partial T}{\partial x} - \frac{1}{ρ} \frac{\partial p}{\partial x} + \frac{p}{ρ^{2}} \frac{\partial ρ}{\partial x}) + v_{y} (c_{p} \frac{\partial T}{\partial y} - \frac{1}{ρ} \frac{\partial p}{\partial y} + \frac{p}{ρ^{2}} \frac{\partial ρ}{\partial y}) + v_{z} (c_{p} \frac{\partial T}{\partial z} - \frac{1}{ρ} \frac{\partial p}{\partial z} + \frac{p}{ρ^{2}} \frac{\partial ρ}{\partial z})) \\ = \frac{\partial}{\partial x} (λ \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} (λ \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (λ \frac{\partial T}{\partial z}) + ρ \dot{q} V + η Ψ - p (\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}) \end{array}$

si49_e

which can be rearranged to

$\begin{array}{l} ρ c_{p} (\frac{\partial T}{\partial t} + \frac{\partial T}{\partial x} + \frac{\partial T}{\partial y} + \frac{\partial T}{\partial z}) \\ = (\frac{\partial p}{\partial t} + v_{x} \frac{\partial p}{\partial x} + v_{y} \frac{\partial p}{\partial y} + v_{z} \frac{\partial p}{\partial z}) + \frac{p}{ρ} (\frac{\partial p}{\partial t} + v_{x} \frac{\partial p}{\partial x} + v_{y} \frac{\partial p}{\partial y} + v_{z} \frac{\partial p}{\partial z} + ρ \frac{\partial v_{x}}{\partial x} + ρ \frac{\partial v_{y}}{\partial y} + ρ \frac{\partial v_{z}}{\partial z}) \\ = (\frac{\partial p}{\partial t} + v_{x} \frac{\partial p}{\partial x} + v_{y} \frac{\partial p}{\partial y} + v_{z} \frac{\partial p}{\partial z}) + \frac{\partial}{\partial x} (λ \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y} (λ \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z} (λ \frac{\partial T}{\partial z}) + ρ \dot{q} V + η Ψ \end{array}$

si50_e (Eq. 12.23)

where we have used the continuity equation (see Eq. 10.16) to set

$\begin{array}{l} \frac{\partial ρ}{\partial t} + v_{x} \frac{\partial ρ}{\partial x} + v_{y} \frac{\partial ρ}{\partial y} + v_{z} \frac{\partial ρ}{\partial z} + ρ \frac{\partial v_{x}}{\partial x} + ρ \frac{\partial v_{y}}{\partial y} + ρ \frac{\partial v_{z}}{\partial z} \\ = & \frac{\partial ρ}{\partial t} + \frac{\partial (ρ v_{x})}{\partial x} + \frac{\partial (ρ v_{y})}{\partial y} + \frac{\partial (ρ v_{z})}{\partial z} \\ = & 0 \end{array}$

si51_e

We have now derived the combined equation for the thermodynamic state and the conservation of energy that is given by Eq. 12.23. As can be seen, the equation is now dependent only on the six field variables, i.e., temperature T, density ρ, pressure p, and velocity $\vec{v}$ . Adding the Navier-Stokes equation and the conservation of mass will result in a total of five equations that we can use to solve for the field variables. The sixth equation we can use is the ideal gas equation (see Eq. 6.4).

12.9 Summary

In this section, we derived the energy equation, i.e., the conservation of energy that gives rise to the fifth equation required for solving the field variables. As stated, in the case of incompressible fluids, the density ρ is constant, and therefore we require a total of only five equations. If the flow problem involves compressible fluids, we need one additional equation. This equation is usually the thermodynamic equation of state and is derived directly from the ideal gas law. This completes the derivation of the fundamental equations because we have now the required number of equations for the field variables.

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Table of Contents for Chapter 12: Conservation of Energy: The Energy Equation and the Thermodynamic Equation of State

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12.1 Introduction

12.2 Energy Transfer by Convection

12.3 Heat Flow by Conduction

12.4 Work Flow by Boundary Forces

12.5 Heat Flow by Volume Effects

12.6 Work Flow by Volume Forces

12.7 Balance of Contributions

12.7.1 Simplifications on the Left-Hand Side

12.7.2 Simplifications on the Right-Hand Side

12.7.3 Dissipation Coefficient

12.8 Thermodynamic Equation of State

12.9 Summary

Table of Contents for
Chapter 12: Conservation of Energy: The Energy Equation and the Thermodynamic Equation of State