There are 11 coefficients in the new product model,coe= [c1c2c3c4c5c6c7c8c9c10c11].
c1 to c5 are equivalent to the coefficients in the present product model. c6 andc7 are the weights of the spray temperature and the product temperature respectively for the container temperature in the gap. c8andc9are the weights of the spray temperature and the product temperature respectively for the container temperature during heating. c10 andc11are the weights of the spray temperature and the product temperature respectively for the container temperature during cooling.
The new product model also gives nice results in the two heating zones. c1as function of
∆T is shown on figure 7.4 andc2as function of ∆T can be seen on figure 7.5.
c8 as function of ∆T can be seen on figure 7.6 and on figure 7.7c9 as function of ∆T is
Most of the values ofc8are approximately 0.9 and the values ofc9are approximately 0.1 only some single values differ from these values. The values which differs from the almost constant values are all for the low spray temperature level, but some values from the low spray temperature level are also on the constant level, so there is no dependency on the spray temperature level. c8andc9 does not seem to have a dependency on ∆T.
The product model for heating becomes
Tc,heat(tn) =Tc,heat(tn−1, Ts, Tp, c1(∆T), c8, c9, dt)
Tp,heat(tn) =Tp,heat(tn−1, Tc,−c2(Ts−level)) , (7.2) wherec1(∆T) =A·∆T+B,c2(Ts−level) is constant in each zone, but different from zone to zone andc8 andc9are constants.
Again the model does not coincide very well with the measured product temperature in the two cooling zones so these data sets are inappropriate and can also not be used to test the cooling coefficients in the new product model.
Chapter 8
COMSOL modelling
To describe the flow and the temperature in the product during pasteurization the heat transfer equation coupled with the non-isothermal Navier-Stokes equations are investi-gated.
The expectations to the flow are that it during heating flows up along the side of the container and down in the middle because the product is heated from the side. During cooling it is expected that the flow turns and flows down along the side and up in the middle. Because of this flow it is expected that the temperature increases from the top of the container during heating and that the temperature decreases from the bottom during cooling. These expectations are based on that beer behaves almost like water and when water is above 4℃and is heated the density decreases.
To the investigation of the flow and temperature the program COMSOL Multiphysics is used. COMSOL uses the finite element method to solve the partial differential equations, [8, p312–319, p328–335, p521].
Normally the finite volume method would be used to solve the Navier-Stokes equations but in this case the velocities in the flow are so small that the finite element method can also be used.
8.1 Partial differential equations
The partial differential equations used to the COMSOL model are the general heat transfer equation coupled with the non-isothermal Navier-Stokes equation, [9, p121–139,p151–
166].
The general heat transfer equation through a fluid with both convection and conduction is
δtsρCp
∂T +∇(−k∇T+ρCpu∇T) =Q , (8.1)
k=k(T),ρ=ρ(T) and Cp=Cp(T). u andvare calculated in the Navier-Stokes equa-tions.
The non-isothermal Navier-Stokes equations are
ρ(u∇)u=∇
where p is the pressure, η is the dynamic viscosity, κ is the dilatational viscosity and F = [FrFz] is the volume force, whereFr is the volume force in ther−direction andFz
is the volume force in thez−direction, [3, ch11,p1–10]. ηis a function of the temperature η=η(T),κ= 0,Fr= 0 andFz= 9.81·(ρ(T0)−ρ(T)) which is the gravity force and has effect as a volume force. When ρ increases the gravity force will increase in downward direction.
The initial conditions for the partial differential equations are T(t0) = T0, u(t0) = 0, v(t0) = 0,p(t0) = 0. The boundary conditions for the heat transfer are different depending on the place on the boundary. On the sides and the top of the container a Dirichlet boundary conditions is used. The boundary condition is a time depending temperature T =T b(t) which is equivalent to the spray temperature. In the bottom of the container the boundary is insulated and the boundary condition is−n(−k∇T+ρCpuT) = 0 which is a Neumann boundary condition. The bottom is insulated because the container during pasteurization stands on a plastic belt conveyor and this means that the bottom is not very much affected by the surrounding temperature. The boundary conditions for the Navier-Stokes equations on all boundaries are that the fluid’s velocity equals the velocity of the boundaries which is 0. This is also called no slip and the boundary condition is u=0which is a Dirichlet boundary condition.
The expressions for k(T), ρ(T),Cp(T) andη(T) are found from the values in table 8.1, [10].
T[K] ρ[mkg3] Cp[kgKJ ] η[P as] k[mKW ] 273.15 999.84 4217.6 0.001793 0.5610 283.15 999.70 4192.1 0.001307 0.5800 293.15 998.21 4181.8 0.001002 0.5984 303.15 995.65 4178.4 0.0007977 0.6154 313.15 992.22 4178.5 0.0006532 0.6305 323.15 988.03 4180.6 0.0005470 0.6435 333.15 983.20 4184.3 0.0004665 0.6543 343.15 977.78 4189.5 0.0004040 0.6631 353.15 971.82 4196.3 0.0003544 0.6700 363.15 965.35 4205.0 0.0003145 0.6753 373.15 958.40 4215.9 0.0002188 0.6791 Table 8.1: Values ofρ,Cp,ηandkfor different values ofT.
Partial differential equations Section 8.1
The values in the table is for water. MATLAB’s fitting tools are used to find cubic expressions for each of the variables as function of the temperature. The cubic expressions are
ρ(T) = 1.569·10−5T3−0.018774T2+ 6.7647T + 233.17 , (8.4) Cp(T) =−1.5227·10−4T3+ 0.16194T2−56.582T+ 10691 , (8.5) η(T) =−2.9827·10−9T3+ 3.0756·10−6T2−0.0010615T + 0.12302 , (8.6) k(T) =−6.6628·10−9T3−2.9149·10−6T2+ 0.0051701T −0.49841 . (8.7) On figure 8.1 to figure 8.4 the values and the cubic expressions for each variable as functions of the temperature can be seen. The figures also show the residuals for each expression. The residuals for each expression are small compared to the size of the variable so the expressions are fine models for the variables.
270 280 290 300 310 320 330 340 350 360 370 380
−0.1
270 280 290 300 310 320 330 340 350 360 370 380 950
Figure 8.1: ρ as function of T and the residuals.
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−2 0 2
residuals
270 280 290 300 310 320 330 340 350 360 370 380 4180
270 280 290 300 310 320 330 340 350 360 370 380
−4
−2 0 2
x 10−5 residuals
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Figure 8.3: η as function of T and the residuals.
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−5 0 5
x 10−4 residuals
270 280 290 300 310 320 330 340 350 360 370 380 0.58
Figure 8.4: k as function ofT and the residuals.