#### 2.1. Calculation of Nucleate Boiling Convective Coefficient

When heat is gradually transferred to the crude oil, several processes are taking place. As the temperature of the crude oil exceeds the boiling point, it begins to evaporate.

At first, individual bubbles form at the heating surface (saturation state). When they reach a particular diameter, they are breaking away and as a result, there is intensive agitation of the boundary layer and the whole mass of boiling liquid. As the heat flux increases the number of crude oil bubbles formed at the heating surface increases. This leads to a further increase in turbulence, and a rapid rise in the heat transfer coefficient. The boiling regime occurring during bubble generation in the vicinity of the tube wall surface is called “nucleate boiling”.

In this boiling regime, high values of convective heat transfer are obtained with relatively small temperature gradient. Nucleate boiling is intensified as the wall temperature increases but is suppressed as the fluid velocity increases. As the forced convection component increases, the nucleate boiling component decreases (by suppression). The most widely used empirical equation which combines these two processes has been derived by Chen [

17]. It takes the form:

Here

${h}_{B}$ denotes the two-phase convective boiling heat transfer coefficient (W/m

^{2} K),

${h}_{FC}$ denotes the forced convective boiling heat transfer coefficient (W/m

^{2} K), and

${h}_{NB}$ is the nucleate boing convective.

${h}_{FZ}$ denotes the nucleate boiling convective heat transfer coefficient (W/m

^{2} K). It is calculated by using the Forster-Zuber empirical equation (see Equation (7)). This correlation gives good results when used in Equation (1).

${S}_{NB}$ is the suppression factor which account for the reduction in the coefficient due to the suppression of nucleation sites. It is related to the total Reynolds number as shown in

Figure 4 [

18].

The suppression factor

S_{NB} is a function of a two-phase Reynolds number, Re

_{TP}:

The two-phase Reynolds number is calculated according to the following equation:

Here

${\mathrm{Re}}_{L}$ denotes the liquid phase Reynolds number. It is defined by:

${F}_{c}$ denotes the Reynolds number factor. This factor accounts for the enhanced flow and turbulence due to the presence of vapor. This term can be approximated by [

18]:

${F}_{c}$ is graphically presented in

Figure 5 [

18].

${X}_{tt}$ denotes the Martinelli parameter. This term is calculated by the following equation [

18]:

The correlation of Forster and Zuber (1955) for the nucleate boiling coefficient has the form [

19]:

where

$\Delta {T}_{sat}$ is the wall superheat (K). It is the wall temperature minus saturation temperature.

$\Delta {p}_{sat}$ is the difference in saturation pressure (N/m

^{2}) corresponding to the temperature difference

$\Delta {T}_{sat}$.

${k}_{L}$ is the thermal conductivity of the liquid (W/m K).

${c}_{pL}$ is the heat capacity of the liquid phase (J/kg K).

${\rho}_{L}$ is the density of the liquid phase (kg/m

^{3}).

${\rho}_{G}$ is the density of the vapor phase (kg/m

^{3}).

${\mu}_{L}$ is the viscosity of the liquid phase (kg/m sec).

$\sigma $ is the surface tension (N/m).

$\Delta {h}_{fg}$ is the enthalpy of vaporization (J/kg). The forced convective coefficient is derived from the Dittus-Boelter relationship [

18]:

The nucleate boiling convective coefficient h

_{NB} is related to the convective coefficient calculated for the mixture properties (

h_{NBM}) as follows [

15]:

where

y denotes the vapor mole fraction of the lighter component equilibrium with the liquid mole fraction of the component

x (see

Figure 6). The mole fractions

x and

y can be obtained by Raoult’s rule. This rule states that the partial pressure of a component over a solution is the product of the vapor pressure of that component and the mole fraction of this component [

20]. The parameter

A_{s} is calculated by the following equation:

Here

p denotes the system pressure in bars.

Ao denotes a constant which depends on the binary system. Stephan and Körner give a table for some combinations. They recommend a value of this term should be 1.5 if no other information is available. This method for binary mixtures has been extended to multicomponent mixtures by Stephan and Preusser [

21].

The crude oil refinery is composed of several unit operations components, such as: reboilers, heat exchanger, and pumps. Analysis and design of the performance of these components require knowledge of the thermodynamic and physical properties of the crude oil fluids [

22]. As described in the previous section designing a reboiler requires knowledge of thermodynamic and thermo-physical properties crude oil. It will be shown later that these are: boiling temperature, pressure, heat capacity, thermal conductivity, diffusivity, and density. Heat exchanger design depends on enthalpies, thermal conductivities, and viscosity of the flowing streams (the crude oil and the steam).

#### 2.2. Estimation of the Thermo-Physical Properties of the Crude Oil

Due to the complexity of the composition crude oils, it is not possible to measure or calculate accurately all of these properties [

22]. Furthermore, calculation methods developed for pure hydrocarbons are not always applicable. Over the years, the petroleum refining engineers have developed empirical correlations in order to estimate the properties of crude oil fraction by measuring properties such as specific gravity (SG) and normal boiling point (NBP). Another important term connected with SG is called API gravity. This term is a measure of the relative density of a crude oil liquid and the density of water (i.e., how heavy or light a crude oil liquid is compared to water) [

23]. These correlations require minimum input data, and are usually presented in graphical form. These quantities are described in the following section.

In order to calculate the nucleate boiling heat flux, it is essential to estimate the thermo-physical properties of the Crude oil. These properties are described in this section. The thermal conductivity of the crude oil as a function of the temperature is shown in

Figure 7 [

24].

The density of liquid crude oil is shown in

Figure 8.

The viscosity of liquid crude oil is shown in

Figure 9.

The latent heat of the crude oil is shown In

Figure 10 [

25].

The surface tension of the crude oil at three different temperatures: 15.56 °C, 37.78 °C, and 54.44 °C as a function of API (see

Figure 11) [

26].

It can be seen from this Figure that the surface tension decreases with the API and the temperature. The viscosity of vapor crude oil is shown in

Figure 12.

Riazi and Al-Sahhaf [

27] evaluated different crude oil properties such as the boiling point, density, refractive index, critical temperature, pressure, and density as a function of the molecular weight by using the following general equation:

Here

$\theta $ can be any one of the properties mentioned above.

M is the molecular weight and

${\theta}_{\infty}$ is the limiting value for any property as

$M\to \infty $. This empirical equation can be applied in order to calculate the following properties.

${T}_{b}$ is the mean average boiling point in

K,

SG is the specific gravity and

${p}_{c}$ is the critical pressure in bar. The constants

a,

b, and

c for each property are given in

Table 1 [

22,

27].

#### 2.3. Fire Dynamic Simulation (FDS) Modeling of the Fired Heater

The FDS has been developed at the National Institutes of Standards and Technology (NIST) [

28,

29]. It solves simultaneously the classical conservation equations, which includes the momentum, energy, mass species equations and thermodynamics-based state equation of a perfect gas. The chemical combustion reaction data are taken from a library of different fuel sources. By solving these equations, it calculates the velocity, pressure by solving the momentum equation), temperature (by solving the Energy equation), density (by solving the Equation of state), and chemical composition (by solving the Diffusion equation) within each numerical grid cell as a function of time. It is capable to calculate the heat flux. FDS is composed of three major components:

Hydrodynamic Model—it is formulated based on of fire-driven fluid flow (i.e., natural convection). The numerical solution is carried out using Large Eddy Simulation (LES). This technique is applied for relatively low speeds (or low Reynolds number).

Combustion Model—two types of combustion models are employed in FDS software. The first (regarded as the default) employs the mixture fraction, a quantity representing the fuel and the products of combustion. The second model makes use of the specified Arrhenius reaction parameters for individual gas species. The mass fractions of all of the major reactants and products can be calculated from the mixture fraction by using means of “state relations”, empirical correlations obtained by a combination of simplified analysis and measurements [

30].

Radiation Transport—radiative heat transfer is obtained by solving of the radiation transport equation (RTE) for a non-scattering gray gas. In a several number of cases, a wide band model can be applied in place of the gray gas model. A Finite Volume Method (FVM) technique is applied in FDS software in order to solve this equation [

29].

The postprocessor of this software is named “smoke-view”. As we shall see in

Section 3, it is capable to perform post-processing images of the temperature, velocity, and soot fields. This code has been employed in this algorithm in order to simulate the Heptane combustion performance.

#### 2.4. Governing Transport Equations of FDS Software

The following subsections describe the conservation momentum, mass (continuity), diffusion, and energy transport equations for the Newtonian fluid.

#### 2.4.1. Mass and Species Transport Equations

Mass conservation is expressed as a function of the density,

$\rho $ [

28]:

where

$\rho $ represents the density of the fluid [kg/m

^{3}], and

**u** is the velocity field [m/s]. This equation is expressed as a function of each gaseous species,

${Y}_{\alpha}$:

where

${D}_{\alpha}$ represents the diffusion coefficient of

$\alpha $ component of the mixture [m

^{2}/s].

#### 2.4.2. Momentum Transport Equation

The momentum equation is described in the following equation [

28]:

Here

**f**_{b} is the force term (gravity force) [Pa/m].

${\tau}_{ij}$ is the stress tensor, [Pa]. This term is defined as [

28]:

The term S_{ij} represents the symmetric rate of strain tensor. μ denotes the dynamic viscosity of the fluid. In this work the Heptane fire heater simulation has been carried out by applying LES method, in which the large-scale eddies are calculated directly, and the subgrid-scale dissipative processes are modeled. FDS uses LES in most fire simulation applications. The following subsection contains a description of how the transport dissipative terms are modeled in FDS software.

#### 2.4.3. Large Eddy Simulation (LES) Turbulence Method

Large Eddy Simulation (LES) is applied in FDS software in order to model the dissipative terms (such as: viscosity, thermal conductivity, and material diffusivity) which occur at length scales smaller than those that are explicitly resolved on the numerical computational grid cell. They are replaced by surrogate expressions that “model” their impact on the approximate form of the governing equations (It means that these properties cannot be applied directly in most practical simulations). The dissipation rate,

$\epsilon $, [Pa/s] is the rate at which kinetic energy is converted to thermal energy by the viscosity of the gaseous mixture [

28]:

The viscosity

μ is computed according to the following equation

Here Δ is a length on the order of the size of a grid cell and

C_{s} represents the empirical constant. The bar above the quantities appearing in Equation (18) represents resolved values. They are calculated from the numerical solution sampled on a coarse grid cell (relative to Direct Numerical Simulation—DNS). The other diffusive parameters such as: material diffusivity and thermal conductivity are functions of the turbulent viscosity. They are calculated by using the following equations [

28]:

It is assumed that the two terms, turbulent Schmidt number $S{c}_{t}$ (defined as the ratio of momentum diffusivity to mass diffusivity) and turbulent Prandtl number $P{r}_{t}$ (defined as the ratio of momentum diffusivity to thermal diffusivity), are constant for a given scenario. The model for the viscosity, ${\mu}_{LES}$, serves two roles: first, it has the appropriate mathematical form to describe the dissipation of kinetic energy from the flow. Second, it provides a stabilizing effect in the numerical algorithm, damping out numerical instabilities as they arise in the flow field, especially where vorticity is generated.

#### 2.4.4. Energy Transport Equation

The energy transport equation is described as a function of the sensible enthalpy,

${h}_{s}$ [J/kg] [

28]:

This term depends on the temperature (according to the ideal law assumption) [

28]:

The sensible heat of each component in the gaseous mixture is calculated by applying Equation (22) [

28]:

Here

${\dot{q}}^{\u2034}$ is the volumetric heat release rate produced by the Heptane combustion [W/m

^{3}],

${\dot{q}}_{b}^{\u2034}$ is the energy transferred to the evaporating Heptane liquid [W/m

^{3}] and

${\dot{q}}^{\u2033}$ is the conductive and radiative heat fluxes [W/m

^{2}] [

28]:

#### 2.4.5. Equation of State

The pressure is computed by applying the ideal gas equation of state [

28]:

where

T is the temperature in [K],

$\overline{W}$ is the molecular mass of the gaseous mixture in [J/mole],

$\overline{R}$ is the gas constant, and

p is the pressure in [Pa].

#### 2.4.6. Fire Dynamics Simulation (FDS) Modeling of the Heptane Burner

The geometric model of the Heptane burner is shown in

Figure 13.

The height of the burner model is 2.4 m. It should be noted that most tube failures in fired heaters are caused by high tube metal temperatures (TMTs). Flame impingement on the tube external surface caused by long flames results in higher tube metal temperatures and coking that affect the performance of the heater and decrease the strength of the tubes (such as thermal creep) [

31]. The length and the width of the burner are 1.2 m. The mesh size of the burner is: 27,468 cells. At the bottom of the burner, liquid Heptane is injected and ignited. The green points indicate the positions of the thermocouples and the soot particle sensors. The heat of combustion of Heptane is 44,600 (kJ/kg) [

32]. The thermal conductivity of the liquid Heptane is 0.14 (W/(m·K)) [

33]. Its specific heat is 2.24 (kJ/(kg·K)) [

34], and its density is 679.5 (kg/m

^{3}) [

35]. The soot yield of Heptane is: 0.037 [

36].

Initial condition—it is assumed that the initial temperature, the component concentration in the air and pressure are:

Boundary Condition—it is assumed that all four boundaries are opened to the atmosphere.

#### 2.6. Convective Heat Transfer across Banks of Tubes

This section deals with heat transfer to bundle of tubes inside the reboiler. For flue gaseous mixture flowing across tube bundles composed of 10 or more rows, Grimison [

37] has obtained a correlation of the form:

where

$N{u}_{D}$ is the Nusselt number. It is defined in the following equation:

where

h is the convection coefficient in [W/(m

^{2}·K)],

D is the tube diameter in [m], and

k is the thermal conductivity of the gaseous mixture in [W/(m·K)]. Pr is the Prandtl number. It is the ratio between the kinematic viscosity, ν [m

^{2}/sec] to the thermal diffusivity, α [m

^{2}/sec]. It is assumed that longitudinal spacing between tube centers (

S_{L}) is equal to the transverse spacing between tube centers (

S_{T}). The spacing between the tube centers is equal to twice of the tube diameter. Thus:

According to Incropera et al. [

38], the values of the constants

C_{1} and

m are:

${\mathrm{Re}}_{D,\mathrm{max}}$ denotes the Reynolds number of the external flow. This term is defined in Equation (29):

D denotes the tube diameter in [m] and

$\nu $ denotes the kinematic viscosity of the flue gaseous mixture.

${V}_{\mathrm{max}}$ [m/sec] is the maximal velocity. It is calculated from Equation (30):

The value of the flue gas velocity,

V, is obtained from FDS calculation of flame. The required transport properties of the flue gaseous mixture were calculated by using STANJAN software [

39]. The required mole fraction of the products species of the oxidation reaction were calculated from the following reaction:

The flame temperature of the Heptane flame was obtained by using the FDS software. The transport thermo-physical properties are listed in

Table 2.

#### 2.7. Calcaulation of the Radiative Heat Flux Emitted by the Mixture Flue Gaseous and the Soot Particles

The of radiative heat transfer within the fire heater enclosure are composed of two components: the high temperature burned gaseous mixture and the soot particles within the Heptane flame [

13]. The burned gaseous mixture is composed mainly of the carbon dioxide and water vapor. They are not grey but emit and absorb radiation only in the discrete bands in the spectrum and therefore, have mean absorption coefficients which vary with gas temperature and the radiation source. The emissivity of the gaseous mixture can be obtained by adding the absorption coefficient for the gas and soot in each spectral line. The overall emissivity of the gas-soot mixture is given in the following equation [

40]:

where:

The weighting factors and absorption coefficients for a three-term equation have been evaluated for the case of a mixture of the products of oil combustion.

${C}_{soot}$ is the soot concentration. It is obtained by applying FDS software (see Figure 23). The weighting factors,

${a}_{g,n}$ varies linearly with temperature, such that:

The values of the coefficients

${b}_{1,n},\text{}{b}_{2,n},\text{}{k}_{s,n},\text{}{k}_{g,n}$ obtained for the mixture emissivity are given in

Table 3. With these values, Equations (32–34) define the gas and flame emissivity in an oil fired furnace where the radiation path length is in the range of 0.2 m to 6 m, and the temperature is in the range of 1100 to 1800 K.