Climate control of a bulk storage room for foodstuffs

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1 Climate control of a bulk storage room for foodstuffs S. an Mourik, H. Zwart, Twente Uniersity, K.J. Keesman*, Wageningen Uniersity, The Netherlands Corresponding author: S. an Mourik Department of Applied Mathematics Uniersity of Twente, The Netherlands Phone: +3 ( , Fa: +3 ( s.anmourik@ewi.utwente.nl * Systems & Control Group Wageningen Uniersity, The Netherlands Karel.Keesman@wur.nl Abstract A storage room contains a bulk of agricultural products, such as potatoes, onions, fruits, etcetera. The products produce heat due to respiration, see for eample [?,?]. A entilator blows cooled air around to keep the products at a steady temperature and preent spoilage. The aim is to design a control law such that the product temperature is kept at a constant, desired leel. The system contains nonlinear coupled partial differential equations (pde s. The assumptions that the input switches between discrete alues, and that the system states hae either ery slow or ery fast dynamics, simplify the equations. Transfer functions of the linear subsystems gie us an idea of the time scales of the states, and with their approimations we can conert linear pde s into ordinary differential equations (ode s. The control problem for the simplified system consists of the determination of the switching moment, and is relatiely easily soled. The storage room We consider a closed storage room with a bulk of products, Figure. The air temperature Figure : Bulk storage room T a (,t is regulated by a entilator that blows the air through the shaft and through the bulk. A cooling element, with ariable temperature T c (t, is placed right below the entilator. This means that if the entilator blows harder, the air is cooled down more. The speed of the entilator and the temperature of the cooling element are the control input. The aim is now to design a control law such that the product temperature, the controlled ariable, is kept at a constant, desired leel.

2 . The model The following assumptions are made: The walls are perfectly insulated; The air temperature in the shaft, T 0 (t, is well-mied and therefore uniform oer the length of the shaft; The air- and product temperature in the bulk, T a and T p, only ary with the height of the bulk, and so not with the width; Diffusion in the air is neglected since conection will dominate the heat transport; No moisture is incorporated in the model; The products are spherical; There is no heat echange between products, but only between the products and the air. These assumptions lead to the following energy balance. The energy inflow (J/s of the air in the entilator shaft is modelled in the basic way: ρcφ(αt c (t + ( αt in (t, with ρ the air density, c the heat capacity, and Φ the olume flow of air through the shaft. The dimensionless constant α denotes the effectieness of the cooling deice; α = implies that the incoming air, T in (t, is totally cooled down (or heated up to the temperature of the cooling element, T c (t, while α = 0 implies that the incoming air is not cooled at all. We assume that α is independent of Φ. Because of the perfect insulation, we hae that T in (t = T a (L,t. The energy outflow equals ρcφt 0 (t. The dynamic energy balance for T 0 (t therefore equals ρcv T 0(t t = ρcφα(t a (L,t T c (t + ρcφt a (L,t ρcφt 0 (t (J/s, ( with V the olume of the shaft. The energy balance for T a (,t is, with (0,L, ρcv 2 T a (,t t with boundary condition = V 2 ρc T a(,t + h(a p (T p (R,,t T a (,t (J/s m 3, (2 T a (0,t = T 0 (t. (3 Here, V 2 is the air olume per bulk olume, A p the product surface area per bulk olume, and T p (R,,t is the product surface temperature at. The two r.h.s. terms in (2 denote the conection of heat and the heat echange between product surface and air respectiely. The parameter h depends on ia the implicit relation (see [?] Nu = (0.5Re / Re 2/3 Pr /3 for 0 < Re < 0 4, (4 with Nu, Re and Pr the Nusselt-, Reynolds- and Prandtl number, see the appendi. The heat transport inside a product at height is modelled by diffusion in a 3D sphere with radius R. T p (r,,t ρ p c p = λ T p(r,,t t r 2 r (r2 + ρ p at p (r,,t + ρ p b (J/m 3 s, (5 r where ρ p,c p, and λ are the product density, heat capacity and conductiity, respectiely. The last two terms in (5 denote the heat production, which is modelled according to [?]. The boundary conditions are T p (,0,t = 0 by symmetry at the origin (6 r λ T p r (,R,t = h((t a(,t T p (R,,t heat flu through the surface. We impose the following additional assumptions 2

3 b 0, because typically at p b. This simplifies the analysis in the net section; The product has no skin; T p (r,,t is regarded as a continuous medium in. The heat flu between air and product surface at is modelled as the flu between a sphere with a uniform air temperature T a (,t along its surface; The last assumption can be justified by showing that the consequent maimal error in the transfer function from T p (R,,s to T a (,s is small. The equations are nonlinear since enters equation ( by Φ = A f, with A f the surface of the bulk floor, and equation (2 ia equation (4. The control objectie is to regulate T p, and the control ariables are and T c..2 Conclusions from the model The equations are nonlinear and coupled. Because of their comple form, a control law cannot be obtained adequately by standard control design, such as e.g. optimal control. Fortunately, for constant (t, the equations are linear. Therefore, we can compute the transfer functions that connect the air- and product temperatures in the frequency domain. So we constrain (t and T c (t to only discrete alues. The control law now consists of determining the optimal switching time instances and the choice of the size of the time interal in which and/or T c are switched. 2 System analysis We want to take three system states, T p (R,L,t, T a (L,t and T 0 (t. The transfer functions that connect these states in the frequency domain can tell us something about the timescales of their dynamics. Also, these transfer functions can be approimated by simple rational ones. First, the orders of the time constants of the uncoupled subsystems are determined. We emphasize that the analysis on the uncoupled subsystems only gies a global impression on the orders of the timescales. The interaction between the coupled states may change the time constants considerably. To be sure, we check the outcome on the full, coupled system. The physical parameters that are used, are for potatoes and are listed in Table. 2. Time constants of the product The heat transfer between the air and the product surface is in the frequency domain T p (R,,s = G p (s T a (,s. (7 The transfer function G p (s is a transcendental function and is approimated by a rational one in order to obtain its time constant. This is done with a Padé[0,] approimation in s = 0 (see appendi a G p (s a 2 + a 3 s. (8 Transformation to the time domain gies T p (R,,t t = a 2 a 3 T p (R,,t + a a 3 T a (,t, (9 3

4 with a = Bi (0 a 2 = 2M 3 cot(m Bi a 3 = R2 cot 2 (M 3 + R2 M 3 cot(m 3. M M M 2 M = λ ; diffusion coefficient (m 2 /s ρ p c p M 2 = a c p ; reaction constant (/s M 3 = chemical reaction rate M 2 /M R; diffusie heat transfer rate. The dimensionless parameter M 3 is analogous to the Thiele modulus chemical reaction rate Th = diffusie mass transfer rate. ( Bi is the Biot number (see appendi and Bi 2. The time constant of G p (s is a 3 /a 2 and is O(0 3. The time constant for heat transfer from bulk air to within the sphere is een higher, because Bi 0.. This means that the heat transport from the product surface to the core takes a long time (see [?], chapter 4.2. We conclude that T p has slow dynamics. 2.2 Time constants of the air temperatures The time constants of T 0 are obtained from the transfer functions that correspond to equation ( T 0 (s = G 3 (s T a (L,s + G 4 (s T c (s (2 = α α + V Ta Φ s (L,s + + V Tc Φ s (s. The time constants are both equal to V Φ. Therefore, the time constant of this subsystem equals V Φ. The transfer function corresponding to equation (2 is T a (,s = 0 M 4 ep ( ( s + M4 z T p (R,z,sdz ep with M 4 = hap V 2ρc. This epression is not of the form s + M 4 ( + T 0 (sep s + M 4, (3 T a (,s = G (s, T 0 (s + G 2 (s, T p (R,,s, (4 and therefore we make an approimation by taking T p (R,,t uniform in. The corresponding transfer functions are now ( G (s, = ep s + M 4 (5 G 2 (s, = M 4( ep ( s+m4 s + M 4. They are not rational since they contain a time delay /. Hence, we cannot see their time constants directly. The Padé[0,] approimations in s = 0 are gien by G (s, G 2 (s, ep ( M 4 ep ( M 4 + ep ( M 4 2(cosh( M4 + M 4 (ep ( M 4 4 s (6 M 4 s,

5 with M 4 = hap V 2ρc the reaction constant of the product heat production (/s. The approimations hae time constants ( at most O(0 0 in = L (7 M 4 (ep ( M 4 M 4 ep ( M 4 ( at most O(0 0 in = L. We hae that the dimensionless number M 5 = M 4L ; is analogous to the Damköhler I number 2.3 Coupled dynamics Da I = chemical reaction rate conectie heat transfer rate (8 chemical reaction rate conectie mass transfer rate. (9 The interaction between the states complicates the analysis. Howeer, we try to derie some global properties of the coupled system in an ad-hoc manner. The starting point is that T a (L,t is constant, and that T c (t switches. Then T 0 (t quickly settles to a new alue because of the small time constant of G 4 (s in (2. We split up (4 into T a (L,s = G (s,l T 0 (s + G 2 (L,s T p (R,L,s (20 = T a (L,s + T a2 (L,s. T a (L,t will settle quickly because of the small time constant of G (s,l. T a2 (L,t adapts itself quickly to T p (R,L,t, but since T p (R,L,t moes slowly (because of the large time constant of G p (s, T a2 (L,t will also moe slowly. T a (L,t is dominated by T a2 (L,t since G 2 (s,l G (s,l s ir. Hence, after a switch, the fast change of T a (L,t will be small. This justifies the starting-point that T a (L,t is constant. If is switched, then we obsere the following. The order of the time constants remain the same if and 2 are of the same order. If increases, then the static gain G (0,L in (5 and (6 increases, whereas G 2 (0,L decreases. So T a (L,t will be dominated less by T p (R,L,t than before, and more by T 0 (t. As a result, T a (L,t will cool down. T a (L,t will settle quickly because G (s,l and G 2 (s,l hae small time constants. 2.4 Full system in state space form We looked at the time constants of the uncoupled subsystems, with the assumption that T p is uniform in. The analysis of the time constants is checked numerically on the (coupled full system. The full system consists of equations (, (2, and (9. We look at the full system dynamics by means of eigenalue analysis and time simulation. The equations are discretised as follows. The term Ta Ta,n Ta,n in (2 is upwind discretised like δ, where the second subscript denotes the discrete space. The spatially discretised total system has the form T t = A(T + B(T c, (2 with T = [T a,..t a,n T 0 T p,..t p,n ] T. Subscript denotes = 0, and n denotes = L. This differential equation is simulated in Simulink with the ode45 Dormand-Prince algorithm. The physical parameters of the chosen configuration are listed in Table. For these constants, Re =.9 0 3, so we may use equation (4. The inerse eigenalues of A( of the full (coupled system are compared to the time constants of the uncoupled subsystems. Table 2 shows that for n = 2 the orders of the time constants of the total system match those of the uncoupled systems. For larger n, the 5

6 α 0.5 T ini 288 R V 0 L λ 0.55 ρ p 04 a A p 49 V c p T c 276 A f 5 Table : Physical parameters of a bulk with potatoes. The data specific for potato were taken from [?,?]. subsystems full system 2.26e3 8.43e3-3.00e i i Table 2: Inerse eigenalues of the full system and time constants of the three uncoupled subsystems. resemblance becomes better, but a difference will remain, because we compare dynamics of states T a (L,t,T 0 (t, and T p (R,t to the spatially distributed state ector [T a,..t a,n T 0 T p,..t p,n ] T. The time constants of the full system contain an imaginary part. This is probably due to the high condition number of A( of O(0 7. Time simulations therefore show oscillations, but when the state ectors of T p, T 0 and T a are simulated in parallel, the oscillations do not show. The (real parts of the eigenectors of the full system are shown in Table 3. The two left eigenectors correspond to the small eigenalues and thus to the large timescales. These are the only ones that hae substantial components in the directions T p, (t and T p,2 (t. So T p has only slow dynamics. This is in agreement with the analysis of the uncoupled subsystems from the preious section. The two slow eigenectors also hae substantial components in the other directions, so the fast states T 0 and T a also hae slow dynamics. We conclude that the fast state dynamics are coupled to the slow ones; they settle quickly and slowly moe along with the slow states. This also in agreement with the preious section. From the inerse eigenalues we see that the time constant of the slow states are O(0 3, and the time constants of the fast states are O(0 0. The switching interal should therefore be chosen in the order of Time simulation The initial product- and air temperatures are 288 K, and we let T c switch between 276 K and 248 K eery fie minutes. Total simulation time is 20 hours. Figure 2 shows T p,n (t for n = 2,0 and 20. For reasonable accuracy we need n = 0, which comes to 2 states in total. Figure 3 shows T p,n (t and T a,n (t on an interal of 0 minutes at the moment of steepest descent of T p,n (t. T a,n (t settles quickly after a switch, whereafter it moes slowly along with T p. After a switch, T 0 (t and T c (t (both not shown here change considerably, whereas T a (L,t changes only a little. T a, T a, T T p, e e e-4 T p, e e e-2 Table 3: State space ector (most left and the eigenectors of the full system for n = 2. 6

7 n=2 n=0 n= Figure 2: Simulation of the cooling of T p,n (t for n = 2,0 and 20. Hence, the influence of T 0 (t on T a (L,t is ery small, in agreement with the analysis from the preious section. Further, the states are all more or less constant during one switching interal T (K T a T p Time (seconds Figure 3: The steepest descent of T p,n (t and T a,n (t during the cooling process. 3 Switching time Because T p has its highest alues at the top of the bulk, we want to control T p (R,L,t. Suppose T p (R,L,t is at its ideal alue T p,opt (R,L. The question is now when to switch T c or (or both such that T p (R,L,t maintains its ideal alue. In the preious section it was shown that on a timescale of 5 minutes, T p (R,L,t and T a (L,t are approimately constant due to the fast settling of T a (L,t and the slow dynamics of T p (R,L,t. T a (L can therefore be computed from the static equations ( and (2. This gies T 0 = ( αt a (L + αt c (22 L 0 T a (L = ep ( M 4 M4 T p(d + αt c. ep(m 5 + α 7

8 Note that in order to sole for (22, one needs to know the spatial distribution of T p (R,. The optimal air temperature is computed from the static equation (9, which gies The switching time, such that the aerage T a (L is optimal, τ 0 T a,opt (L = a 2 a 3 T p,opt (L. (23 τf T a, (Ldt + T a,2 (Ldt = τ f T a,opt (L, (24 τ where subscripts and 2 denote the inputs,t c, and 2,T c,2, has the solution τ = τ f(t a,opt T a,2 (L T a, (L T a,2 (L τ = τ f(t a,opt T a, (L T a,2 (L T a, (L if (,T c = (,T c, at t = 0 (25 if (,T c = ( 2,T c,2 at t = 0. 4 Conclusion We hae modelled the storage room by one ordinary differential equation and two partial differential equations. The resulting system equations are simplified using three techniques; timescale separation, discrete switching input, and Padé approimation of a transfer function. The transfer functions of the simplified system gie a good indication of the timescales, and also show how the fast dynamics are coupled to the slow dynamics. These properties are checked by analysis of the discretized full system. For the simplified system, the switching time τ is completely parametrized. This gies an algorithm for keeping T p (R,L,t at a constant alue. The methodology used here seems etendable to more comple models which incorporate, for eample, moisture transport and heat loss through the walls. The net step will be the construction of a controller to steer T p (R,L,t. The input will be T a (L,t. The discrete time steps, in which the controller prescribes the optimal input, are then equal to τ f. By continuous adjustment of τ, the aerage T a (L oer the switching interal is set to its prescribed alue. 5 Acknowledgement This work was supported by the Technology foundation STW under project number WWI Appendi A. Construction of a transfer function A transfer function of a linear pde in the ariables and t is constructed by substituting / t = s and soling the ode for the ariable. The solution can be written as ŷ(s, = G(s,û(s,, with ŷ(s,, û(s, and G(s, the output, input and the transfer function, respectiely. If G(s, is of the form a 0 s a n s n b 0 s b m s m, (26 i.e. rational, then the transfer function for a fied can be transformed back into a linear ode in t, and its time constant can be determined. If not, the nonrational transfer function can be approimated by a rational one, for eample by a Padé approimation. 8

9 A.2 Padé approimation A Padé[0,] approimation in s = 0 has the form and the coefficients are determined by setting G(s, = a 2 a ( + a3 a (s, (27 G(0, = G(0, G G (0, = s s (0,. A cleer choice of n and m in a Padé[n,m] approimation is made by obseration of the Bode plot of the original transfer function. A.3 Notation α cooling effectieness T p produce temperature (K α th thermal diffusiity of air (.87e-5 m 2 /s T a air temperature in the bulk (K λ conduction of product (W/m K T c cooling element temperature (K λ air conduction of air (2.43e-2 W/m K τ f length of switching interal (s ρ p produce density (kg/m 3 T ini initial temperature (K ρ air density (.27 kg/m 3 V olume of shaft (m 3 τ switching time (s V 2 porosity (m 3 /m 3 ν kinematic iscosity of air (.3465e-5 m 2 /s c p heat capacity of produce (J/kg K a product heat production (J/kg s K c heat capacity of air (000 J/kg K b product heat production (J/kg s h heat transfer coefficient (W/m 2 K R product radius (m air elocity through bulk (m/s A p produce surface per bulk olume (m 2 /m 3 Φ air flow through shaft (m 3 /s A f floor area of the bulk (m 2 M λ ρ pc p a L bulk height (m M 2 c p L 2 R por( por, char. length (m M 3 M2 /M R Bi Biot number 2hR ha λ M p 4 V 2ρc Re Nu M 4L Reynolds number L2 ν, see [?] M 5 Nusselt number 2hR ν λ air Pr Prandtl number α th 9