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TKP4145
Reactor technology
Project 2
Lise Jensen, Kasper Linnestad, Anders Leirpoll and Kjetil Sonerud
March 25, 2014
Contents
1 Objectives 4
2 Differential Equations 42.1 Species Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Ideal Gas Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Numerical method 7
4 MATLAB functions 84.1 Mole Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Average Molar Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Ideal Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.5 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.6 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.7 Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.8 Heat Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.9 Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.10 Derivative of Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.11 Derivative of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.12 Derivative of Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.13 Derivative of Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.14 Derivative of Mass Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.15 Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.16 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.17 Inlet States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.18 Mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 MATLAB main script 285.1 Main . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Results 30
List of program codes
4.1 getMolarFractions.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 getAvgMolarMass.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 getIdealDensity.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 getViscosity.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.5 getReynolds.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.6 getHeatCapacity.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.7 getReaction.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.8 getHeatCoefficients.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.9 getDiffusivity.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.10 getPressureDerivative.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.11 getTemperatureDerivative.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.12 getDensityDerivative.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.13 getVelocityDerivative.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.14 getMassFractionDerivative.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2
4.15 reactor.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.16 getConstants.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.17 getInletStates.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.18 getMassMatrix.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1 main.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3
1 Objectives
In this project, the synthesis gas production process in a multi-tube reactor is studied:
Steam reforming:CH4 + H2O −−⇀↽−− CO + 3 H2 (1.1)
Water-gas-shift-reaction:CO + H2O −−⇀↽−− CO2 + H2 (1.2)
Sum of Eq. (1.1) and (1.2):CH4 + 2 H2O −−⇀↽−− CO2 + 4 H2 (1.3)
The reaction is carried out in a multi-tube packed bed reactor. One of the tubes is modelled in detail totake a closer look at the pressure, temperature and velocity profiles, as well as the mole fractions of thecomponents.
2 Differential Equations
The differential equations for mass, energy and momentum are given below. In addition, the continuityequation (from the total mass balance) and the differential equation for density (from the ideal gas law) isused. These are solved using a numerical method as discussed in Sec. 3, and solved in MATLAB as discussedin Sec. 4 and Sec. 5.
Due to the geometrical properties of the reactor, cylindrical coordinates are used in all the differentialequations.
2.1 Species Mass Balance
The complete species mass balance is given in Eq. 2.1 below. This equation is used to derive an expression
for∂ω
∂z.
∂ρω
∂t+∇ · (ρωuT) = −∇ · j + R (2.1)
where the terms represent the transient term, convection, dispersion and reaction rate, respectively.
By assuming steady-state, the transient term can be neglected. The radial convective contribution and theaxial dispersion can be neglected, as well as any change in angular direction due to symmetry. The reactionrate is modified so that it is expressed in terms of moles. The dispersion term in the radial direction is givenby
jr = −ρD ∂ω
∂r(2.2)
By applying the aforementioned assumptions, Eq. (2.1) reduces to
∂
∂z(ρωuz) =
1
r
∂
∂r
(rρD
∂ω
∂r
)+ R (2.3)
where ρ is the density of the gas, r is the radius of the tube, D is the mass transfer coefficients, ω is themass fractions, uz is the axial velocity and R is the reaction rates on a mass basis.
4
Applying the product rule and solving for ∂ω∂z
∂ω
∂z=
1
ρuz
−ωuz ∂ρ∂z− ρω ∂uz
∂z+ D
(ρ
r
∂ω
∂r+ ρ
∂2ω
∂r2+∂ω
∂r
∂ρ
∂r
)+ R
(2.4)
where
R = (1− ε)ρcatMw ⊗NT r (2.5)
ρcatis the density of the catalyst, ε is the void fraction, Mmass is the molar masses, NT is the stoichiometricmatrix and r are the reaction rates of the different reactions occurring in the reactor on a molar basis.
2.2 Ideal Gas Law
Want to find an expression for how the density of the gas changes down the reactor. The density is givenby the ideal gas law
ρ =Mmassp
RT(2.6)
where Mmass is the average molar mass of the gas and p is the pressure. Differentiating the expression for ρwith respect to the axial length of the reactor, z, assuming that the average mole weight does not change inthe axial direction.
∂ρ
∂z=
∂
∂z
(Mmassp
RT
)=Mmass
R
(1
T
∂p
∂z− p
T 2
∂T
∂z
)(2.7)
2.3 The Equation of Continuity
The equation of continuity is used to derive an expression for the derivative of the velocity with respect toz. The equation for is given by
∂ρ
∂t+∇ · (ρu) = 0 (2.8)
Assuming steady state ( ∂∂t = 0), symmetry in angular direction ( ∂∂θ = 0), and that the convection in theradial direction is negligible, the continuity equation is reduced to
∂ρuz∂z
= 0 (2.9)
∂uz∂z
= −uzρ
∂ρ
∂z(2.10)
2.4 Energy Balance
The energy balance is used to derive an expression for ∂T∂z . The energy balance, neglecting the heat of
mixing, pV -work and the viscous dissipation term is given by
ρcp∂T
∂t+ ρcpu
T · ∇T = ∇ · (λ∇T )−∆rxHTR (2.11)
Where the terms represent the transient term, convective term, diffusion term and the reaction term, respec-tively. Considering steady state and symmetry in angular direction, and assuming that the diffusion in axial
5
direction, the convective contribution in radial direction and that the change in effective radial conductionis negligible, the energy balance becomes
∂T
∂z=
1
ρcpuz
λeff,r
(1
r
∂T
∂r+∂2T
∂r2
)−∆rxHTR
(2.12)
2.5 Momentum Balance
To estimate the pressure drop in the reactor, Ergun’s equation is applied [4]
∂p
∂z= −fρgu
2
dt(2.13)
where ∂p∂z is the derivative of pressure with respect to z, f is the friction factor, ρg is the cross-sectional
averaged gas density, u is the cross-sectional averaged fluid velocity and dt is the tube diameter.
The friction factor is given in Eq. 2.14 as a function of void fraction, ε, and the cross-sectional averagedReynold’s number of the particle, NRe.
f =1− εε3
(1.75 + 4.2N
5/6Re
1− εNRe
)(2.14)
6
3 Numerical method
In this project, the MATLAB solver ode15s is used to solve the differantial equations [3]. The solver is basedon numerical differentiation formulas and optionally the backward differentiation formulas (also known asGear’s method). It is a multi-step solver—in computing y(tn), it makes use of preceding time points i,y(tn−i). In radial direction dss020 and dss042 are used to discretize the differential equations of first andsecond order respectively. These files utilize a higher order finite difference scheme. ode15s is used tointegrate in axial direction. To satisfy the boundary conditions, a mass matrix is applied.
In general, ode45 is regarded by MathWorks (the company behind MATLAB) as “(...)the best function toapply as a first try for most problems.”[2]. However, since the current problem is stiff, ode15s is applied.This solver is also used in the case of DAEs. A stiff problem will consist of several processes, at least one ofwhich will have a small time constant. The stiff nature of the differential equations can clearly be observed inthe temperature plot, where the drop at the inlet is near instantaneous. This is also experienced by runningthe ode45 solver, which would use hours to solve the problem, whereas ode15s has a runtime of 4 s, with asatisfying amount of discrete points.
In this project we have set the relative tolerance and the absolute tolerance to 108 and 1010 respectively. Wealso increased the number of radial discretization points to 30 to achieve a higher resolution for the figures.
z[m]
r[m]
Figure 3.1: The reactor tube is discretized in axial direction using ode15s and radial direction using dss020
and dss042 for first and second order derivatives, respectively. The points containing boundary conditionsat r = 0 and r = R are shown in bright red, while the the remaining points are shown in faded red.
7
4 MATLAB functions
4.1 Mole Fractions
Program code 4.1: getMolarFractions.m1 function y = getMolarFractions(omega, mMass)
2 %% Converts mass fractions to molar fractions
3 % Input:
4 % omega: mass fractions ω5 % Matrix: Columns correspond to ri, rows to components.
6 % mMass: molecular mass Mmass
7 % Vector: Each row correspond to each component
8 % Output:
9 % y: mole fractions y10 % Matrix: Same as ω11 y = zeros(size(omega));
12 for i=1:size(omega,2)
13 y(:,i) = (omega(:,i)./mMass)/(omega(:,i)’*(1./mMass));
14 end
15 end
4.2 Average Molar Mass
Program code 4.2: getAvgMolarMass.m1 function avgmolarmass = getAvgMolarMass(molarMass, molefraction)
2 %% Average Molar Mass
3 % Calculates the average molar mass
4 % Input:
5 % molarMass: The molecular weights, Mmass [kg mol−1]6 % molefraction: Mole fractions of the components, y7 % Output:
8 % avgmolarmass: The average molar mass, Mmass [kg mol−1]9
10 % Mmass = MTmassy
11 avgmolarmass = molarMass’*molefraction; % [kg mol−1]12 end
8
4.3 Ideal Density
Program code 4.3: getIdealDensity.m1 function rho = getIdealDensity(pressure, AvgMolarMass, temperature)
2 %% Ideal density
3 % Calculates the density of an ideal gas
4 % Input:
5 % pressure: The total pressure, p, [Pa]6 % AvgMolarMass: The average molecular weight, Mmass [kg mol−1]7 % temperature: The temperature, T [K]8 % Output:
9 % rho: The density, ρ [kg m−3]10
11 % The gas constant, R12 r = 8.3144621; % [J mol−1K−1]13
14 % ρ = MmasspRT
15 rho = AvgMolarMass.*pressure./(r.*temperature); % [kg m−3]16 end
4.4 Viscosity
Program code 4.4: getViscosity.m1 function mu = getViscosity(temperature, S, B, molefraction)
2 %% Viscosity
3 % Calculates the viscosities of the components at a given temperature
4 % and mole fraction
5 % Input:
6 % temperature: The temperature, T [K]7 % S: Component viscosity coefficient [K]8 % B: Component viscosity coefficient [kg K0.5ms−1]9 % molefraction: The molefractions (optional)
10 % Output:
11 % mu: The viscosity or the average viscosity depending on the number of
12 % inputs.
13 % µi = biT1.5
T+Si
14 % µ = µTy15
16 mu = zeros(length(S),length(temperature));
17 for i = 1:length(S)
18 mu(i,:) = (B(i)*temperature.^1.5)./(temperature+S(i));
19 end
20 if nargin == 4
21 % The molefractions are given, calculate the average viscosity
22 muMatrix = mu;
23 mu = zeros(size(temperature));
24 for i = 1:length(temperature)
25 mu(i) = muMatrix(:,i)’*molefraction(:,i);
26 end
27 end
28 end
9
4.5 Reynolds Number
Program code 4.5: getReynolds.m1 function re = getReynolds(density, velocity, diameter, viscosity)
2 %% Reynold’s number
3 % Calculates the Reynolds Number for flow in a pipe or tube, using the fluid
4 % density, flow velocity, characteristic diameter and dynamic fluid
5 % viscosity
6 % Input:
7 % density: The density of the gas, ρg [kg m−3]8 % velocity: The velocity of the gas, uz [m s−1]9 % diameter: The diameter of the particles, dp [m]
10 % viscosity: The dynamic viscosity of the gas, µ [kg s−1m−1]11 % Output:
12 % re: The Reynold’s number, NRe
13
14 % NRe = udρµ
15 re = density.*velocity.*diameter./viscosity;
16 end
4.6 Heat Capacity
Program code 4.6: getHeatCapacity.m1 function cp = getHeatCapacity(temperature, coeffMatrix, molefraction)
2 %% Heat Capacity
3 % Calculates the heat capacities of the components at a given temperature,
4 % and returns them as a column vector.
5 % Input:
6 % temperature: The temperature, T [K]7 % coeffMatrix: The coefficients α, β, γ, δ as a
8 % matrix
9 % molefraction: The molefractions (optional)
10 % Output:
11 % cp: The heat capacities
12 % cp = α + β T + γ T 2 + δ T 3
13 % cp = yTcp14
15 cp = coeffMatrix*[ones(size(temperature)); temperature; temperature.^2; temperature.^3];
16 % [J mol−1K−1]17 if nargin == 3
18 % The molefractions are given, calculate the average heat capacity
19 cpMatrix = cp;
20 cp = zeros(size(temperature));
21 for i = 1:length(temperature)
22 cp(i) = cpMatrix(:,i)’*molefraction(:,i);
23 end
24 end
25 end
10
4.7 Reaction
This file was given with the problem formulation, but it has been modified somewhat.
Program code 4.7: getReaction.m1 function [rrx, deltaH] = getReaction(T, ...
2 y, ...
3 P, ...
4 Aj, ...
5 actEn, ...
6 Ax, ...
7 adEnt, ...
8 ent298, ...
9 ent948)
10 %% Reaction
11 % This function calculates the reaction rates for all the components
12 % and the heat of the reaction in all the discretication points.
13 % Input:
14 % T [=] K Temperature
15 % y [=] - Mol fraction vector
16 % P [=] Pa Total pressure
17 % Aj [=] kmol/kgcat h Preexponential factors as a vector
18 % actEn [=] J/kmol Activation energies for the reactions as a vector
19 % gasConst [=] J/kmol K Universal Gas constant
20 % Ax Preexponential factors for the adsorbtion constants
21 % adEnt [=] J/kmole Activation energies for the reactions
22 % ent298 [=] J/kmol Reaction enthalpies at 298K
23 % ent948 [=] J/kmol Reaction enthalpies at 948K
24 %
25 % Output
26 % rrx [=] mol/kg(cat)s Reaction rates for each reaction
27 % (#r-points x #components-matrix)
28 % deltaH [=] J/m^3s Reaction heat for each reaction
29 % (#r-points x #components-matrix)
30
31 %% Initializing
32 gasConst = 8.3144621; % Gas constant
33 numr = size(T,2); % Number of components
34 nComp = size(y,1); % Number of radial discretization points
35
36 %% Reaction enthalpies
37 % Three reactions for each point in the r-direction
38 deltaH = zeros(3,numr);
39 for i=1:3
40 deltaH(i,:)=ent298(i)*ones(1,numr)+(T-298*ones(1,numr))./(948-298).*(ent948(i)-ent298(i));
41 end
42
43 %% Partial pressures
44 pComp = zeros(nComp,numr);
45 for i=1:nComp
46 pComp(i,:) = y(i,:).*P/1e5;
47 end
48
49 % Initialization: reaction rate for each point in the r-direction
11
50 rrx = zeros(3,numr);
51 % Initialization: denominator in the rate expression for each point in
52 % the r-direction
53 denom = zeros(1,numr);
54
55 for i=1:numr
56 % Rate constant
57 Krx = Aj.*exp(-actEn./(gasConst*T(1,i)));
58
59 % Adsorbtion constant
60 Kads = Ax.*exp(-adEnt./(gasConst*T(1,i)));
61
62 % Equilibrium constants
63 Keq(1) = 10^(-11650/T(1,i)+13.076);
64 Keq(2) = 10^(1910/T(1,i)-1.784);
65 Keq(3) = Keq(1)*Keq(2);
66
67 % Reaction rates
68 denom(i) = 1 + Kads(2)*pComp(2,i) + Kads(3)*pComp(4,i) ...
69 + Kads(1)*pComp(1,i) + Kads(4)*pComp(5,i)/pComp(4,i);
70
71 rrx(1,i)= Krx(1)/(pComp(4,i))^2.5*(pComp(1,i)*pComp(5,i) ...
72 - (pComp(4,i))^3*pComp(2,i)/Keq(1))/(denom(i))^2;
73
74 rrx(2,i)= Krx(2)/(pComp(4,i))*(pComp(2,i)*pComp(5,i) ...
75 - (pComp(4,i))*pComp(3,i)/Keq(2))/(denom(i))^2;
76
77 rrx(3,i)= Krx(3)/(pComp(4,i))^3.5*(pComp(1,i)*(pComp(5,i))^2 ...
78 - (pComp(4,i))^4*pComp(3,i)/Keq(3))/(denom(i))^2;
79 end
80
81 end
4.8 Heat Coefficients
his file was given with the problem formulation, but it has been modified somewhat.
Program code 4.8: getHeatCoefficients.m1 function [Ur,lambdaer]=gethHeatCoefficients(Re,...
2 T,...
3 Y,...
4 gasViscosity,...
5 cpGas,...
6 particleDiameter,...
7 radiusi,...
8 radiuso,...
9 lambdaSt,...
10 epsilon,...
11 lambda,...
12 avgMolarMass)
13 % heatcoef
14 % The function computes the heat transfer coefficient for radial transport of
15 % heat from the bed to the surroundings
16 % Input:
12
17 % Re [=] - Reynolds number
18 % T [=] K Temperature
19 % Y [=] - Mol fraction
20 % VIS [=] kg/ms Gas viscosity
21 % CPgas [=] J/kgK Gas heat capasity
22 %
23 % Output
24 % Ur [=] J/m^2sK Heat coefficient
25 % LAMBDAer [=] J/msK Effective radial conductivity
26
27 %% Constants
28 numr = size(Y,2);
29 %nComp = size(Y,1);
30 pconst = 1.0;
31 beta = 1.0;
32 lambdas = 0.243;
33 phi = 0.3;
34
35 % Mass based heat capacity
36 cpGas = cpGas./avgMolarMass;
37
38 %Calculates the gas heat conductivity
39 Tmatrix = [ones(size(T)); T; T.^2; T.^3];
40 lambdacomp = lambda*Tmatrix;
41 lambdag = diag(Y’*lambdacomp)’;
42
43 %Prandtl number
44 Pr = gasViscosity.*cpGas./lambdag;
45
46 %Radial effective static conduction
47 alpharv = (0.227/(1+epsilon/(2*(1-epsilon))*(1-pconst)/pconst)*(T/100).^3);
48 alphars = 0.227*pconst/(2-pconst)*(T/100).^3;
49 lambdaer0 = lambdag.*(epsilon*(1 + beta*particleDiameter*alpharv./lambdag) + ...
50 beta*(1-epsilon)./(1./(1/phi + alphars*particleDiameter./lambdag) + 2/3*lambdag/lambdas));
51
52 %Effective radial conductivity
53 lambdaer = lambdaer0+0.14*lambdag.*Re.*Pr;
54
55 %Heat transfer coefficient near the wall
56 alphaw0=8.694/(2*radiusi)^(4/3)*lambdaer0(numr);
57 alphaw=alphaw0+0.444*Re(numr)*Pr(numr)*lambdag(numr)/particleDiameter;
58
59 %Overall heat transfer coefficient
60 Ur=(radiusi*log(radiuso/radiusi)/lambdaSt+1/alphaw)^(-1);
61
62 end
13
4.9 Diffusivity
Program code 4.9: getDiffusivity.m1 function diffusivity = getDiffusivity(dparticle,velocity,radiusi)
2 %% Calculates the diffusivity
3 % The Peclet numbers given in the problem formulation
4 % Input:
5 % dparticle: Diameter of the particles dp
6 % velocity: The axial velocity at each radial discretization uz7 % radiusi: The inner radius of the tubes ri
8 % Output:
9 % diffusivity: The diffusivity D [m2 s−1]10
11 peNumberrd = 8*(2-(1-2*dparticle/radiusi)^2); % Peclet number
12 peNumbermr = 1.1*peNumberrd; % Peclet number
13 diffusivity = velocity*dparticle/peNumbermr;
14 end
14
4.10 Derivative of Pressure
Program code 4.10: getPressureDerivative.m1 function dpdz = getPressureDerivative(rho,u,re,void,dp,r)
2 %% Derivative of pressure
3 % This function computes the 1D pressure gradient in the system by using
4 % superficial velocity, density and Reynolds number.
5 % Input:
6 % rho: Gas density ρ [kg m−3]7 % Row vector of r
8 % u: Velocity uz [m s−1]9 % Row vector of r
10 % re Reynolds number
11 % Row vector of r
12 % void: Void fraction ε13 % Row vector of r
14 % dp: Particle diameter dp
15 % Constant
16 % r: The radius r17 % Vector
18 % Output:
19 % dpdz Pressure gradient∂p
∂z20
21 % Cross-sectional averaging
22 format long
23 rho = trapz(r,rho.*r)/(.5*r(end)^2);
24 u = trapz(r,u.*r)/(.5*r(end)^2);
25 re = trapz(r,re.*r)/(.5*r(end)^2);
26 % Friction factor, f = 1−εε3 (1.75 + 4.2Re(5/6)(1−ε)
re )27 f = (1-void)/void^3*(1.75+4.2*re^(5/6)*(1-void)/re);
28 % The Ergun equation,∂p
∂z=
−fρ u2z
dp
29 dpdz = -f*rho*u^2/dp*ones(size(rho));
30 end
15
4.11 Derivative of Temperature
Program code 4.11: getTemperatureDerivative.m1 function dT = getTemperatureDerivative(rho,...
2 heatCapacity,...
3 velocity,...
4 lambda,...
5 radius,...
6 rhoCat,...
7 reactionRate,...
8 reactionEnthalpy,...
9 dtdr,...
10 dt2dr2,...
11 void,...
12 avgMolarMass)
13 %% Temperature Derivative
14 % Calculates the derivative of the temperature
15 % Input:
16 % temperature: The temperature, T [K]17 % rho: The density of the gas, ρ [kg m−3]18 % heatCapacity: The heat capacity of the gas, cp [J mol−1K−1]19 % velocity: The velocity of the gas, uz [m s−1]20 % lambda: The effective radial conductivity, λr, [J s−1m−1K−1]21 % radius: The inner tube radius, r [m]22 % rhoCat: The density of the catalyst, ρcat [kg m−3]23 % reactionRate: The rate of reaction, r [mol s−1kg−1]24 % reactionEnthalpy: The enthalpy of reaction, ∆rxH [J mol−1]
25 % dtdr: The result from dss020.m,∂T
∂r
26 % dt2dr2: The result from dss042.m,∂2T
∂r2
27 % void: The void fraction ε28 % avgMolarMass: The average molar mass Mmass [kg mol−1]29 % Output:
30 % dT: The derivative of the temperature,∂T
∂z31
32 heatCapacity = heatCapacity./avgMolarMass;
33 % Total enthalpy of reactions at each radial point
34 dH = zeros(size(rho));
35 for i = 1:length(rho)
36 dH(i) = reactionRate(:,i)’*reactionEnthalpy(:,i);
37 end
38 %(1-void)*rhoCat*dH
39 %∂T
∂z= (ρ cp uz)
−1
[λr
r (∂T
∂r+ r
∂2T
∂r2) + (rρcat(−∆rxH)
]40 dT = ((rho.*heatCapacity.*velocity).^-1) .*...
41 (lambda.*(1./(radius+1e-50).*dtdr + dt2dr2) -...
42 (1-void)*rhoCat*dH); % [K m−1]43 end
16
4.12 Derivative of Density
Program code 4.12: getDensityDerivative.m1 function drho = getDensityDerivative(temperature, pressure, averageMolarMass, pressureDerivative, temperatureDerivative)
2 %% Density Derivative
3 % Calculates the change in density in z-direction in a multitube reactor
4 %
5 % Input:
6 % temperature: The temperature, T [K]7 % Vector: columns correspond to ri8 % pressure: The pressure, p [Pa]9 % Vector: columns correspond to ri
10 % averageMolarMass: The average molar mass, M_w [g/mol]
11 % Vector: columns correspond to ri
12 % pressureDerivative: The derivative of the pressure,∂p
∂z[Pa m−1]
13 % Vector: columns correspond to ri
14 % temperatureDerivative: The derivative of the temperature∂T
∂z[K m−1]
15 % Vector: columns correspond to ri16 % Output:
17 % drho: The derivative of the density,∂ρ
∂z[kg m−4]
18
19 % The gas constant, R20 r = 8.3144621; % [J mol−1K−1]21
22 %∂ρ
∂z= Mmass
R
(1T
∂p
∂z− p
T 2
∂T
∂z
)23 drho = averageMolarMass/r.*(1./temperature.*pressureDerivative - ...
24 pressure./temperature.^2 .* temperatureDerivative);
25 end
26
27
17
4.13 Derivative of Velocity
Program code 4.13: getVelocityDerivative.m1 function du = getVelocityDerivative(velocity,...
2 density,...
3 densityDerivative)
4 %% Velocity Derivative
5 % Calculates the derivative of the velocity
6 % Input:
7 % velocity: The velocity of the gas, uz [m s−1]8 % density: The gas density, ρ, [kg m3]
9 % densityDerivative: The derivative of the gas density,dρ
dz[kg m−4]
10 % Output:
11 % du: The derivative of the velocity,∂uz∂z
12
13 %∂uz∂z
= −uz
ρ
∂ρ
∂z14 du = -velocity.*densityDerivative./density; % [s−1]15 end
4.14 Derivative of Mass Fraction
Program code 4.14: getMassFractionDerivative.m1 function domega = getMassFractionDerivative(rho,...
2 rhoCat,...
3 velocity,...
4 diffusivity,...
5 radius,...
6 reactionRate,...
7 densityDerivative,...
8 velocityDerivative,...
9 omega,...
10 domegadr,...
11 domega2dr2,...
12 drhodr,...
13 void,...
14 mMass,...
15 stoMat)
16 %% Weight fraction derivative
17 % Calculates the derivative of the weight fraction with respect to z.
18 % All the inputs are row vectors where the columns correspond to each
19 % discretization point of the radius.
20 % Input:
21 % rho: The density of the gas ρ [kg m−3]22 % Row vector
23 % rhoCat: The density of the catalyst, ρcat [kg m−3]24 % Constant
25 % velocity: The velocity of the gas, uz [m s−1]26 % Row vector
27 % diffusivity: The diffusivity D [m s−1]28 % Column vector corresponding to each component
18
29 % radius: The radius of the reactor tube r [m]30 % Row vector
31 % reactionRate: The reaction rate r [mol s−1kg−1]32 % Matrix: rows correspond to reactions
33 % desityDerivative: The derivative of the density with respect to z∂ρ
∂z[kg m−4]
34 % Row vector
35 % velocityDerivative: The derivative of the velocity with respect to z∂uz∂z
[m s−2]
36 % Row vector
37 % omega: The mass fractions ω38 % Matrix: rows correspond to the components
39 % domegadr: The derivative of the weight fraction with respect to r∂ω
∂r[m−1]
40 % Matrix: rows correspond to the components
41 % domega2dr2: The second derivative of the weight fraction with respect to r∂2ω
∂r2[m−2]
42 % Matrix: rows correspond to the components
43 % drhodr: The derivative of the density∂ρ
∂r[kg m−4]
44 % Row vector
45 % void: The void fraction ε46 % Constant
47 % mMass: Molar mass Mmass [kg mol−1]48 % Column vector corresponding to each component
49 % stoMat: Stoichiometric matrix N
50 % Each row correspond to each reaction, each column
51 % correspond to each component
52 % Output
53 % domega: The derivative of the weight fraction with respect to z∂ω
∂z[m−1]
54 % Matrix: rows correspond to the components
55
56 % Reaction rate for each component at each radial point
57 reactionRateComp = zeros(size(omega));
58 for i = 1:length(rho)
59 reactionRateComp(:,i) = (1-void)*rhoCat*mMass.*(stoMat’*reactionRate(:,i));
60 end
61 domega = zeros(size(omega));
62 if size(diffusivity,1)==1
63 % Use overall diffusion (not dependent on component)
64 for i = 1:size(omega,1)
65 domega(i,:) = 1./(rho.*velocity) .* (...
66 -omega(i,:).*velocity.*densityDerivative -...
67 rho.*omega(i,:).*velocityDerivative +...
68 diffusivity .* (rho./(radius+1e-50).*domegadr(i,:) +...
69 rho.*domega2dr2(i,:) +...
70 domegadr(i,:).*drhodr) +...
71 reactionRateComp(i,:));
72 end
73 else
74 % Diffusion is dependent on each component
75 error(’Not yet implemented’);
76 end
19
77 end
4.15 Reactor
Program code 4.15: reactor.m1 function dx = reactor(~,x,par)
2 %% The reactor model
3 % To be used in a matlab integrator (ode45, ode15s, etc.)
4 % Calculates the derivative of the mass fractions, temperatures, velocities
5 % and pressure, given the mass fractions, temperature, velocity, the partial
6 % pressures at all the radial discretization points, and all the other
7 % parameters.
8 % Input:
9 % x: The states
10 % par: The parameters as a struct
11 % rhoCat: The density of the catalysator, ρcat [kg m−3]12 % nu: The stoichiometric coefficients, ν13 % u: The overall heat transfer coefficient, U [J s−1m−2K−1]14 % r0: The radius of the tube, r0 [m]15 % dp: The diameter of the particles, dp [m]16 % tOutside: The temperature outside, Toutside [K]17 % void: The void fraction, ε18 % viscosity: The dynamic viscosity, µ [kg m−1s−1]19 % mMass: The molar masses, Mmass [kg mol−1]20 %% Extracting parameters
21 stoMat = par.nu; % Stoichiometric matrix
22 molecularMass = par.mMass; % Molar mass [kg mol−1]23 numr = par.numr; % Number of radial discretizations
24 numComp = par.nComp; % Number of chemical components
25 rhoCat = par.rhoCat; % Density of catalyst [kg m−3]26 void = par.eps; % Void fraction
27 dparticle = par.dp; % The diameter of the particles m28 radius = par.radius; % The discretized radius [m]29 s = par.s; % Viscosity coefficients [K]30 b = par.b; % Viscosity coefficients [kg m−1s−1K−.5]31 cpCoefficients = par.cp; % Heat capacity coefficients [J mol−1K−n]32 lambda = par.lambda; % Conductivity coefficients [W m−1K−n]33 rateConstants = par.aj; % Rate constants [mol kg−1s−1]34 adsConstants = par.ax; % Adsorbtion constants [Pa−n]35 actEnergy = par.actEn; % Activation energies [J mol−1]36 adsEnthalpy = par.adEnt; % Adsorption enthalpies [J mol−1]37 enthalpy298 = par.ent298; % Reaction enthalpies [J mol−1]38 enthalpy948 = par.ent948; % Reaction enthalpies [J mol−1]39 radiusi = par.radiusi; % Inner tube radius [m]40 radiuso = par.radiuso; % Outer tube radius [m]41 lambdaTube = par.lambdaSt; % Heat conductivity of the wall [W m−1K−1]42 tempOut = par.tempOut; % Outside temperature [K]43 %% Extracting states
44 % Mass fractions
45 massFrac = zeros(numComp,numr);
46 for i = 1:numComp
47 numMass = (1+(i-1)*numr):i*numr;
48 massFrac(i,:) = x(numMass)’;
49 end
20
50 % Temperature
51 numTemp = (numMass(end)+1):(numMass(end)+numr);
52 temperature = x(numTemp)’;
53 % Pressure
54 numPressure = (numTemp(end)+1):(numTemp(end)+numr);
55 pressure = x(numPressure)’;
56 % Velocity
57 numVelocity = (numPressure(end)+1):(numPressure(end)+numr);
58 velocity = x(numVelocity)’;
59 %% Calculations
60 moleFrac = getMolarFractions(massFrac,molecularMass);
61 avgMolarMass = getAvgMolarMass(molecularMass,moleFrac);
62 rho = getIdealDensity(pressure,avgMolarMass,temperature);
63 viscosity = getViscosity(temperature,s,b,moleFrac);
64 re = getReynolds(rho,velocity,dparticle,viscosity);
65 heatCapacity = getHeatCapacity(temperature,cpCoefficients,moleFrac);
66 [reactionRate, reactionEnthalpy] = getReaction(temperature,moleFrac,...
67 pressure,rateConstants,...
68 actEnergy,adsConstants,...
69 adsEnthalpy,enthalpy298,...
70 enthalpy948);
71 [heatTransfer,lambdaEffective] = getHeatCoefficients(re,temperature,...
72 moleFrac,viscosity,...
73 heatCapacity,dparticle,...
74 radiusi,radiuso,...
75 lambdaTube,void,...
76 lambda,...
77 avgMolarMass);
78 diffusivity = getDiffusivity(dparticle,velocity,radiusi);
79 %% Radial derivatives
80 % Mass fractions
81 dMassFracdr = zeros(size(massFrac));
82 dMassFrac2dr2 = dMassFracdr;
83 for i = 1:numComp
84 dMassFracdr(i,:) = dss020(radius(1),radius(end),massFrac(i,:),1);
85 end
86 for i = 1:numComp
87 dMassFrac2dr2(i,:) = dss042(radius(1),radius(end),...
88 massFrac(i,:),dMassFracdr(i,:),2,2);
89 end
90 % Temperature
91 dTemperaturedr = dss020(radius(1),radius(end),temperature,1);
92 dTemperature2dr2 = dss042(radius(1),radius(end),...
93 temperature,dTemperaturedr,2,2);
94 % Density
95 dRhodr = dss020(radius(1),radius(end),rho,1);
96
97 % Velocity
98 dVelocitydr = dss020(radius(1),radius(end),velocity,1);
99 %% Axial derivatives
100 dPressuredz = getPressureDerivative(rho,velocity,re,void,dparticle,radius);
101 dTemperaturedz = getTemperatureDerivative(rho,...
102 heatCapacity,...
103 velocity,...
21
104 lambdaEffective,...
105 radius,...
106 rhoCat,...
107 reactionRate,...
108 reactionEnthalpy,...
109 dTemperaturedr,...
110 dTemperature2dr2,...
111 void,...
112 avgMolarMass);
113 dRhodz = getDensityDerivative(temperature,pressure,avgMolarMass,dPressuredz,dTemperaturedz);
114 dVelocitydz = getVelocityDerivative(velocity,rho,dRhodz);
115 dMassFracdz = getMassFractionDerivative(rho,...
116 rhoCat,...
117 velocity,...
118 diffusivity,...
119 radius,...
120 reactionRate,...
121 dRhodz,...
122 dVelocitydz,...
123 massFrac,...
124 dMassFracdr,...
125 dMassFrac2dr2,...
126 dRhodr,...
127 void,...
128 molecularMass,...
129 stoMat);
130 %% Construct the vector containing all the state derivatives
131 dx = zeros(size(x));
132 % Mass fractions
133 for i = 1:numComp
134 numMass = (1+(i-1)*numr):i*numr;
135 dx(numMass) = [dMassFracdr(i,1)
136 dMassFracdz(i,2:end-1)’
137 dMassFracdr(i,end)];
138 end
139 numTemp = (numMass(end)+1):(numMass(end)+numr);
140 dx(numTemp) = [dTemperaturedr(1,1)
141 dTemperaturedz(1,2:end-1)’
142 dTemperaturedr(1,end) + ...
143 heatTransfer/lambdaEffective(1,end)*...
144 (temperature(1,end)-tempOut)];
145 % Pressure
146 numPressure = (numTemp(end)+1):(numTemp(end)+numr);
147 dx(numPressure) = dPressuredz’;
148 % Velocity
149 numVelocity = (numPressure(end)+1):(numPressure(end)+numr);
150 dx(numVelocity) = [dVelocitydr(1)
151 dVelocitydz(2:end-1)’
152 dVelocitydr(end)];
153 end
4.16 Constants
Program code 4.16: getConstants.m
22
1 function const = getConstants
2 %% Constants
3 % Sets all constants needed in the calculations as a struct.
4 % Input:
5 % -
6 % Output:
7 % const: Struct containing all the constants needed
8 %Initial data
9
10 %Stochiometric matrix
11 const.species = {’CH4’ ’CO’ ’CO2’ ’H2’ ’H2O’ ’N2’};
12 % CH4 CO CO2 H2 H20 N2
13 const.nu = [-1 1 0 3 -1 0 % rx 1
14 0 -1 1 1 -1 0 % rx 2
15 -1 0 1 4 -2 0]; % rx 3
16 %-------------------------------------------------------------------
17 const.Tin = 793; % Initial temperature [K]
18 const.pin = 29e5; % Initial pressure [Pa]
19 const.uin = 1.89; % Velocity [m/s]
20
21
22 % Constants
23 %-------------------------------------------------------------------
24 const.gasConst = 8.3145; % Gas constant [J/K mol]
25 const.nComp = 6; % Number of components [-]
26 const.numz = 30; % Number of axial discretization points
27 const.numr = 30; % Number of radial discretization points
28 const.mpart = 6; % Number of radial discretization points
29 % in the pellet
30 % Catalyst data
31 %-------------------------------------------------------------------
32 const.rhoCat = 2355.2; % Density catalyst [kgcat/m^3]
33 const.eps = 0.528; % Pore fraction [-]
34 const.dp = 0.0173; % Particle diameter [m]
35 const.rp = const.dp/2; % Particle radius [m]
36 const.hp = 30000; % Heat transfer coefficient [W/m^2K]
37 const.kCat = 50; % Particle conductivity [W/mK]
38 const.av = 3/const.rp.*(1-const.eps); % Particle surface area per volume [1/m]
39
40 % Tube data
41 %-------------------------------------------------------------------
42 const.tempOut = 1100; % Temp. outside the tube [K]
43 const.lambdaSt = 52; % Heat coef. for tube metal[W/mK]
44 const.radiusi = 0.051; % Inner radius of the tube [m]
45 const.radiuso = 0.066; % Outer radius of the tube [m]
46 const.length = 7.0; % Tube length [m]
47 const.radius = linspace(0,const.radiusi,const.numr); % Radial discretization points [m]
48
49 % Initial massfraction of the components [-]
50 %-------------------------------------------------------------------
51 const.fracIn(1,1) = 0.1911; % CH4
52 const.fracIn(2,1) = 0.0001; % CO
53 const.fracIn(3,1) = 0.0200; % CO2
54 const.fracIn(4,1) = 0.0029; % H2
23
55 const.fracIn(5,1) = 0.7218; % H2O
56 const.fracIn(6,1) = 0.0641; % N2
57
58
59 % Molemass of the components [kg/mol]
60 %-------------------------------------------------------------------
61
62 const.mMass(1,1) = 16.04e-3; % Molemass CH4
63 const.mMass(2,1) = 28.01e-3; % Molemass CO
64 const.mMass(3,1) = 44.01e-3; % Molemass CO2
65 const.mMass(4,1) = 2.02e-3; % Molemass H2
66 const.mMass(5,1) = 18.02e-3; % Molemass H2O
67 const.mMass(6,1) = 28.01e-3; % Molemass N2
68
69
70 % Preexponential factors for the rate constants [mol/kgcat s]
71 %-------------------------------------------------------------------
72 const.aj(1,1) = 4.255E15/3.6; % Factor for rx. 1
73 const.aj(1,2) = 1.955E6/3.6; % Factor for rx. 2
74 const.aj(1,3) = 1.020E15/3.6; % Factor for rx. 3
75
76
77 % Preexponential factors for the adsorbtion constants
78 %-------------------------------------------------------------------
79 const.ax(1,1) = 6.65E-9; % Factor for CH4 [Pa^-1]
80 const.ax(1,2) = 8.23E-10; % Factor for CO [Pa^-1]
81 const.ax(1,3) = 6.12E-14; % Factor for H2 [Pa^-1]
82 const.ax(1,4) = 1.77E5; % Factor for H2O [-]
83
84
85 % Activation energies for the reactions [J/mol]
86 %-------------------------------------------------------------------
87 const.actEn(1,1) = 240.1e3; % Activation energy for rx. 1
88 const.actEn(1,2) = 67.13e3; % Activation energy for rx. 2
89 const.actEn(1,3) = 243.9e3; % Activation energy for rx. 3
90
91
92 % Reaction enthalpies at 298K [J/mol]
93 %-------------------------------------------------------------------
94 const.ent298(1,1) = 206.1E3; % Enthalpy for rx. 1
95 const.ent298(1,2) = -41.15E3; % Enthalpy for rx. 2
96 const.ent298(1,3) = 164.9E3; % Enthalpy for rx. 3
97
98
99 % Reaction enthalpies at 948K [J/mol]
100 %-------------------------------------------------------------------
101 const.ent948(1,1) = 224.E3; % Enthalpy for rx. 1
102 const.ent948(1,2) = -37.30E3; % Enthalpy for rx. 2
103 const.ent948(1,3) = 187.5E3; % Enthalpy for rx. 3
104
105
106 % Adsorption enthalpies [J/mol]
107 %-------------------------------------------------------------------
108 const.adEnt(1,1) =-38.28E3; % Enthalpy for adsorption of CH4
24
109 const.adEnt(1,2) =-70.65E3; % Enthalpy for adsorption of CO
110 const.adEnt(1,3) =-82.90E3; % Enthalpy for adsorption of H2
111 const.adEnt(1,4) = 88.68E3; % Enthalpy for adsorption of H2O
112
113
114 % Heat capasity coefficients for the components
115 %-------------------------------------------------------------------
116 CP(1,1) = 1.925E4; % 1. coefficient for CH4 [J/kmoleK]
117 CP(1,2) = 5.213E1; % 2. coefficient for CH4 [J/kmoleK^2]
118 CP(1,3) = 1.197E-2; % 3. coefficient for CH4 [J/kmoleK^3]
119 CP(1,4) =-1.132E-5; % 4. coefficient for CH4 [J/kmoleK^4]
120
121 CP(2,1) = 3.087E4; % 1. coefficient for CO [J/kmoleK]
122 CP(2,2) =-1.285E1; % 2. coefficient for CO [J/kmoleK^2]
123 CP(2,3) = 2.789E-2; % 3. coefficient for CO [J/kmoleK^3]
124 CP(2,4) =-1.272E-5; % 4. coefficient for CO [J/kmoleK^4]
125
126 CP(3,1) = 1.980E4; % 1. coefficient for CO2 [J/kmoleK]
127 CP(3,2) = 7.344E1; % 2. coefficient for CO2 [J/kmoleK^2]
128 CP(3,3) =-5.602E-2; % 3. coefficient for CO2 [J/kmoleK^3]
129 CP(3,4) = 1.715E-5; % 4. coefficient for CO2 [J/kmoleK^4]
130
131 CP(4,1) = 2.714E4; % 1. coefficient for H2 [J/kmoleK]
132 CP(4,2) = 0.9274E1; % 2. coefficient for H2 [J/kmoleK^2]
133 CP(4,3) =-1.381E-2; % 3. coefficient for H2 [J/kmoleK^3]
134 CP(4,4) = 0.7645E-5; % 4. coefficient for H2 [J/kmoleK^4]
135
136 CP(5,1) = 3.224E4; % 1. coefficient for H2O [J/kmoleK]
137 CP(5,2) = 0.1924E1; % 2. coefficient for H2O [J/kmoleK^2]
138 CP(5,3) = 1.055E-2; % 3. coefficient for H2O [J/kmoleK^3]
139 CP(5,4) = 0.3596E-5; % 4. coefficient for H2O [J/kmoleK^4]
140
141 CP(6,1) = 3.115E4; % 1. coefficient for N2 [J/kmoleK]
142 CP(6,2) =-1.357E1; % 2. coefficient for N2 [J/kmoleK^2]
143 CP(6,3) = 2.680E-2; % 3. coefficient for N2 [J/kmoleK^3]
144 CP(6,4) =-1.168E-5; % 4. coefficient for N2 [J/kmoleK^4]
145
146 % Units --> SI
147 const.cp = CP/1000; % [J/molK^4]
148
149
150 % Conductivity coefficients for the components
151 %-------------------------------------------------------------------
152 const.lambda(1,1) =-1.869E-3; % 1. coefficient for CH4 [W/mK]
153 const.lambda(1,2) = 8.727E-5; % 2. coefficient for CH4 [W/mK^2]
154 const.lambda(1,3) = 1.179E-7; % 3. coefficient for CH4 [W/mK^3]
155 const.lambda(1,4) =-3.614E-11; % 4. coefficient for CH4 [W/mK^4]
156
157 const.lambda(2,1) = 5.067E-4; % 1. coefficient for CO [W/mK]
158 const.lambda(2,2) = 9.1025E-5; % 2. coefficient for CO [W/mK^2]
159 const.lambda(2,3) =-3.524E-8; % 3. coefficient for CO [W/mK^3]
160 const.lambda(2,4) = 8.199E-12; % 4. coefficient for CO [W/mK^4]
161
162 const.lambda(3,1) =-7.215E-3; % 1. coefficient for CO2 [W/mK]
25
163 const.lambda(3,2) = 8.015E-5; % 2. coefficient for CO2 [W/mK^2]
164 const.lambda(3,3) = 5.477E-9; % 3. coefficient for CO2 [W/mK^3]
165 const.lambda(3,4) =-1.053E-11; % 4. coefficient for CO2 [W/mK^4]
166
167 const.lambda(4,1) = 8.099E-3; % 1. coefficient for H2 [W/mK]
168 const.lambda(4,2) = 6.689E-4; % 2. coefficient for H2 [W/mK^2]
169 const.lambda(4,3) =-4.158E-7; % 3. coefficient for H2 [W/mK^3]
170 const.lambda(4,4) = 1.562E-10; % 4. coefficient for H2 [W/mK^4]
171
172 const.lambda(5,1) = 7.341E-3; % 1. coefficient for H2O [W/mK]
173 const.lambda(5,2) =-1.013E-5; % 2. coefficient for H2O [W/mK^2]
174 const.lambda(5,3) = 1.801E-7; % 3. coefficient for H2O [W/mK^3]
175 const.lambda(5,4) =-9.100E-11; % 4. coefficient for H2O [W/mK^4]
176
177 const.lambda(6,1) = 3.919E-4; % 1. coefficient for N2 [W/mK]
178 const.lambda(6,2) = 9.966E-5; % 2. coefficient for N2 [W/mK^2]
179 const.lambda(6,3) =-5.067E-8; % 3. coefficient for N2 [W/mK^3]
180 const.lambda(6,4) = 1.504E-11; % 4. coefficient for N2 [W/mK^4]
181
182
183 % Viscosity coefficients
184 %-------------------------------------------------------------------
185 const.b(1,1) = 1.00E-6; % Coefficient for CH4 [kg/msK^0.5]
186 const.b(2,1) = 1.50E-6; % Coefficient for CO [kg/msK^0.5]
187 const.b(3,1) = 1.50E-6; % Coefficient for CO2 [kg/msK^0.5]
188 const.b(4,1) = 0.65E-6; % Coefficient for H2 [kg/msK^0.5]
189 const.b(5,1) = 1.74E-6; % Coefficient for H2O [kg/msK^0.5]
190 const.b(6,1) = 1.40E-6; % Coefficient for N2 [kg/msK^0.5]
191
192 const.s(1,1) = 165; % Coefficient for CH4 [K]
193 const.s(2,1) = 220; % Coefficient for CO [K]
194 const.s(3,1) = 220; % Coefficient for CO2 [K]
195 const.s(4,1) = 67; % Coefficient for H2 [K]
196 const.s(5,1) = 626; % Coefficient for H2O [K]
197 const.s(6,1) = 108; % Coefficient for N2 [K]
198
199
200 % Diffusion volumes
201 %-------------------------------------------------------------------
202 const.sumV(1,1) = 25.14; % Coefficient for CH4 [-]
203 const.sumV(1,2) = 18.01; % Coefficient for CO [-]
204 const.sumV(1,3) = 26.90; % Coefficient for CO2 [-]
205 const.sumV(1,4) = 6.12; % Coefficient for H2 [-]
206 const.sumV(1,5) = 13.10; % Coefficient for H2O [-]
207 const.sumV(1,6) = 18.50; % Coefficient for N2 [-]
208
209 end
26
4.17 Inlet States
Program code 4.17: getInletStates.m1 function x0 = getInletStates(par)
2 %% Transforms the inlet states
3 % Returns the inlet states as vectors with length equal to the number of
4 % radal discretization points.
5 % Input:
6 % par: The parameters given by getConstants
7 % Output:
8 % x0: The inlet states [ω0 T0 p0 uz,0]T
9 % at each radial discretization point. All the boundary conditions
10 % are set to zero.
11 omega0 = par.fracIn; % Mass fractions ω12 t0 = par.Tin; % Inlet temperature T0
13 p0 = par.pin; % Inlet pressure p0
14 u0 = par.uin; % Inlet velocity uz,015
16 numr = par.numr; % Number of radial discretization points
17
18 x0 = kron([omega0;t0;p0;u0],ones(numr,1));
19 end
4.18 Mass matrix
Program code 4.18: getMassMatrix.m1 function m = getMassMatrix(x0,par)
2 %% Creates the mass matrix
3 % Input:
4 % x0: The inlet states at all radial discretization points
5 % [ω0 T0 p0 uz,0]T
6 % par: The parameters given by getConstants.m
7 % Output:
8 % m: The mass matrix to be used in the ode solver
9
10 numr = par.numr;
11 % Initializing the mass matrix
12 m = eye(length(x0));
13 % Setting zeros where we have boundary conditions
14 % Mass fractions and temperature:
15 boundaryStates = length(x0)/numr - 2;
16 for i = 1:boundaryStates
17 m(1+(i-1)*numr,1+(i-1)*numr) = 0;
18 m(i*numr,i*numr) = 0;
19 end
20 % Velocity
21 m(1+(i+1)*numr,1+(i+1)*numr) = 0;
22 m((i+2)*numr,(i+2)*numr) = 0;
23 m = sparse(m);
24 end
27
5 MATLAB main script
5.1 Main
Program code 5.1: main.m1 %% Main file
2 % Authors:
3 % Kjetil Sonerud
4 % Lise Jensen
5 % Anders Leirpoll
6 % Kasper Linnestad
7 % Created: 01-Feb-14
8 % Last edited: 27-Feb-14
9 % Requires:
10 % dss020.m
11 % dss042.m
12 % getAvgMolarMass.m
13 % getConstants.m
14 % getDensityDerivative.m
15 % getDiffusivity.m
16 % getHeatCapacity.m
17 % getHeatCoefficient.m
18 % getIdealDensity.m
19 % getInletStates.m
20 % getMassFractionDerivative.m
21 % getMassMatrix.m
22 % getMolarFractions.m
23 % getPlotMolarFractions.m
24 % getPlotPressure.m
25 % getPlotTemperature.m
26 % getPlotVelocity.m
27 % getPressureDerivative.m
28 % getReaction.m
29 % getReynolds.m
30 % getTemperatureDerivative.m
31 % getVelocityDerivative.m
32 % getViscosity.m
33 % reactor.m
34 % matlabfrag (https://dl.dropboxusercontent.com/u/6304542/matlabfrag.m)
35 % Comments:
36 % This script simulates a Fixed Bed Reactor
37 %% TKP4145 - Reactor technology
38 % Project 2
39 clear all
40 close all
41 clc
42 tic;
43 %% Given information
44 par = getConstants;
45 x0 = getInletStates(par);
46 massMatrix = getMassMatrix(x0,par);
47 %% Calculation
48 options = odeset(’RelTol’,1e-10,’AbsTol’,1e-10,’Mass’,massMatrix);
49 [z, x] = ode15s(@reactor,[0 par.length],x0,options,par);
28
50 %% Calculating run time
51 runtime = toc;
52 fprintf(’The script used %.3f sec.\n’,runtime);
53 %% Extract the states
54 % Radius
55 radius = par.radius;
56 % Mass fractions
57 numComp = par.nComp;
58 numr = par.numr;
59 for i = 1:numComp
60 indexMass = 1+(i-1)*numr:i*numr;
61 massFrac(i,:,:) = x(:,indexMass);
62 end
63 % Temperature
64 indexTemperature = indexMass(end)+1:indexMass(end)+numr;
65 temperature = x(:,indexTemperature);
66 % Pressure
67 indexPressure = indexTemperature(end)+1:indexTemperature(end)+numr;
68 pressure = x(:,indexPressure);
69 % Velocity
70 indexVelocity = indexPressure(end)+1:indexPressure(end)+numr;
71 velocity = x(:,indexVelocity);
72 %% Plot
73 getPlotTemperature(temperature,z,[min(min(temperature)) max(max(temperature))],’’,radius)
74 getPlotPressure(pressure,z,[min(min(pressure)) max(max(pressure))]/1e5,’’,radius)
75 getPlotVelocity(velocity,z,[min(min(velocity)) max(max(velocity))],’’,radius)
76 % Convert mass fractions to molar fractions
77 moleFrac = zeros(size(massFrac));
78 for i = 1:length(z)
79 moleFrac(:,i,:) = getMolarFractions(squeeze(massFrac(:,i,:)),par.mMass);
80 end
81 getPlotMolarFractions(moleFrac,z,par.species,’’,radius)
29
6 Results
Fig. 6.1 - 6.3 display the change in mole fractions of H2O, H2, N2, CH4, CO and CO2 along the length ofthe reactor. As expected, the mole fractions of the reactants, CH4 and H2O, decrease as they are used inthe reaction, and the mole fractions of H2 and CO2 increase. The rate of change is caused by the reactionrate, which is dependent on the reactor temperature. As N2 is an inert component in the reactor, its molefraction remains constant.
Mole fraction of H2O
y H2O
r [m]z [m]
0
2
4
60
0.010.02
0.030.04
0.05
0.4
0.45
0.5
0.55
0.6
0.65
0.7
(a) Mole fraction of H2O.
Mole fraction of CH4
y CH
4
r [m]z [m]
0
2
4
60
0.010.02
0.030.04
0.05
0.05
0.1
0.15
0.2
(b) Mole fraction of CH4.
Figure 6.1: Mole fractions of species; H2O and CH4
Mole fraction of H2
y H2
r [m]z [m]
0
2
4
60
0.010.02
0.030.04
0.05
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(a) Mole fraction of H2.
Mole fraction of CO2
y CO
2
r [m]z [m]
0
2
4
60
0.010.02
0.030.04
0.05
0.01
0.02
0.03
0.04
0.05
0.06
(b) Mole fraction of CO2.
Figure 6.2: Mole fractions of species; H2 and CO2
Fig. 6.4 shows how the pressure change along the length of the reactor. When the gas velocity through thereactor increases, the pressure drops according to Ergun’s equation, (2.13). The linearity of the pressuredrop is due to conservation of mass, ρ0u0 = ρguz, which makes Ergun’s equation linear. Conservation ofmass is not explicitly used in the modeling, but it is observed that this relationship is valid.
The reactor temperature profile can be seen in Fig. 6.5. At the reactor tube inlet the temperature is(relatively) low due to the reaction being endothermic and rate of reaction being high. This is due tothe fact that the reaction rate is a function of concentration, which is high at the inlet. The reactioncontinues to cool down the reactor tube until the lowest temperature point is reached at 1,9 · 10−8 m ofTmin = 843,73 K. From here, the external reactor heating exceeds the heat of reaction and the temperature
30
Mole fraction of N2
y N2
r [m]z [m]
0
2
4
60
0.010.02
0.030.04
0.05
0.032
0.034
0.036
0.038
0.04
(a) Mole fraction of N2.
Mole fraction of CO
y CO
r [m]z [m]
0
2
4
60
0.010.02
0.030.04
0.05
0.01
0.02
0.03
0.04
0.05
0.06
0.07
(b) Mole fraction of CO.
Figure 6.3: Mole fractions of species; N2 and CO
Pressure in the reactor
p[bar]
r [m]z [m]
0
2
4
60
0.010.02
0.030.04
0.05
28.2
28.3
28.4
28.5
28.6
28.7
28.8
28.9
Figure 6.4: Total pressure in the reactor.
increases, until it stabilizes. From Fig. 6.6, it is seen that this directly affects the velocity profile.
The velocity profile in the reactor is shown in Fig. 6.6. At the reactor tube inlet the temperature is low,which causes a decrease in velocity as expected from the assumption of ideal gas. The temperature increasesfurther down the tube, which together with the pressure drop causes the velocity to increase.
31
Temperature in the reactor
T[K
]
r [m]z [m]
0
2
4
60
0.010.02
0.030.04
0.05
750
800
850
900
950
1000
1050
Figure 6.5: Temperature profile in the reactor.
Axial velocity in the reactor
uz[m
s−1]
r [m]z [m]
0
2
4
60
0.010.02
0.030.04
0.05
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
Figure 6.6: Velocity profile in the reactor.
32
References
[1] Jakobsen, Hugo A., Fixed Bed Reactors - lecture notes in TKP4145. Department of Chemical Engineering,NTNU. Spring 2011.
[2] http://www.mathworks.se/help/matlab/ref/ode45.html, retrieved 27.02.2014
[3] http://www.mathworks.se/help/matlab/ref/ode15s.html, retrieved 28.02.2014
[4] Ergun, S., Fluid flow through packed columns, Chemical Engineering Progress, 48 (2), 89-94
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