RH simulation model for canvas paintings protected by an aluminium backplate and an additional hygroscopic layer

The model considers four partial processes of mass exchange (Fig. 2), driven by the differences in moisture concentration: (i) permeation and (ii) infiltration from the room to the void through the canvas, (iii) sorption between the void and the canvas and (iv) sorption between the cotton layer and the void. We assume that other mass transfer processes that may occur in this system are negligible in comparison to these four. In particular, the model assumes that the transference of moisture between the void and the wood of the stretcher is negligible small. This may seem surprising given the high relative mass of the wood compared with other materials, but previous research indicates that this assumption is reasonable. Thybring [17] estimates the water vapour diffusivity in wood ranging from 10⁻¹³ to 10⁻¹¹ m²/s. For textiles, as cotton canvas, the values are from 10⁻⁹ to 10⁻⁸ m²/s from Henry [18].

Another key assumption is that any potential gradients of water concentration within the canvas and cotton do not affect the rate of processes 1, 3 and 4 i.e., that these materials can be modelled with a lumped parameter approach.

Calculation procedure by time steps

The calculation procedure takes place time step by time step. At the time step zero the temperatures and RH do not change, and we calculate the mixing ratios, MR room and MR void, with the initial temperatures and the RH values at the room and the void. We use:

We have chosen to use an equation for psat which is accurate at a wide range of temperatures [19]:

At the start of time step one, the temperature changes in all points (Room, Void, Canvas, Aluminium and Cotton) and thus the RH values change as well. From the initial MR Void value and the new temperature values we calculate the new RH values for the Canvas, Aluminium and Cotton layers at the start of the time step one:

With layer as canv, alum and cott, and using (1a) for psat. We obtain the new MR room with Eqs. (1 and 1a) from the news T room and RH room.

From these values we can find, in the current time step, the increase of mixing ratio (ΔMR) in the void from each partial process by following the calculations for each of them, as described in from “Humidity change at the void from the vapour permeation through the canvas (Partial process 1)” to “Humidity change at the void from the vapour sorption in the linen canvas (Partial process 4)” sections. At the end of each time step we calculate the total increase of mixing ratio in the void (ΔMR void), and thus obtain the new MR void. Once this is known, the time step ends, and we can repeat the calculation procedure for the next time step. This iterating procedure, time step by time step, allows us to calculate the evolution of the system for as long as necessary.

Cycle temperatures calculation for each time step

The time steps used in modelling need to be far smaller than the frequency of measurements of the experimental data. As a result, it is efficient to replace the temperature input data (T canv, T cott, T room and T alum) by values calculated with a harmonic oscillation, which fits perfectly with the data used in this case. For any of the locations of interest:

where β (rad) and ω are fitted to the data of each studied case (Additional file 1). This strategy is possible thanks to the oscillating shape of the input data. To use the model with non-oscillating data, it would be necessary to interpolate the values of T in order to obtain a time-series with time-steps that match the calculation. If the thermal resistance of each layer is known, it is also possible to predict the temperature in every layer given the external temperature. However, this is outside the scope of this paper.

Humidity change at the void from the vapour permeation through the canvas (Partial process 1)

The water vapour flows by permeation through the canvas between the void and the room (first partial process). It depends on the canvas permeance and the RH in both canvas sides. To calculate the MR void perm_i we use:

where Mvap void perm is the mass vapour permeation flow (kg), P is the Permeance (kg/(m² s Pa), A is the area (m²), t is the time (s), and Pvap is the vapour pressure (Pa).

As in this equation the variable RH canv depends on the canvas and room temperatures, which change along the cycles, it shall be solved by a numerical FEM method, integrating in a discrete way by time steps along the cycles, as explained at point 1 and 3.1, with Tcanv_i and Troom_i both fixed at each time step. From Eq. (2) we get Eq. (3) and it is:

where Δt is the time step.

The vapour mass inside the void by permeation at the end of the time step i is:

and

With: MR void perm_i in kg vapour/kg/air; ρ air void_i in kg/m³; Vol void in m³.

The ΔMR void perm_i obtained by the canvas permeation process along this time step is:

Humidity change at the void from the air infiltration flow (Partial process 2)

The vapour flow through the space between canvas and stretcher is the second partial process and it depends on the air infiltration rate (volumes/h in the void) and thus on the air-tightness of the canvas attachment to the stretcher (e.g. tension in the canvas, staples distances between them) and possible holes.

Appendix A.1 shows the development and solution of the differential equation that models the air infiltration rate and we get the Eq. (4), which allows us to calculate the vapour concentration in the void by the air infiltration after time interval ∆t:

With:

• CsA (vapour concentration in the void at the start of time step i) = MR void_i−1.

• CsB (vapour concentration at the end of time step i) = MR void_i.

• Ce (vapour concentration in room air) = MR room_i.

• nVol/h (air infiltration rate at the void in Volumes/h).

• ρ rate (density rate) = ρ room/ρ void.

By the air infiltration process at the end of time step “i” we get MR void inf_i = CsB and:

Humidity change at the void from vapour sorption in the cotton layer (Partial process 3)

The cotton vapour sorption is the third partial process involved and it depends on two factors:

Sorption factor 1: The Cotton Sorption rate (%/min) along the cycle depends on the sorption mode (adsorption or desorption) and on the RH values at air boundary layer facing the cotton (RH cott). Sorption factor 2: The cotton EMC (equilibrium moisture content) limits the amount of water adsorbed in the cotton. It depends on the sorption mode (adsorption or desorption) and on the RH cott.

If the EMC is attained and the void receives more water vapour, it accumulates temporarily inside the void, increasing the void MR void and displacing the RH cott cycle phase from the cotton temperature cycle phase. This fact affects the other partial processes as they depend also on the void RH values.

To deal with factor 1 we have fitted one equation to the adsorption rate and other equation to the desorption rate, both describe a relationship between Cotton Sorption rate (%/min) and RH. The values used come from the averaged values at five points in the 4 graphics for Dark/Brown Cotton in Ceylan et al. [20 Fig. 4, lines 3, 6, 9 and 12]. See our Fig. 3a and b.

From these averaged values, in the range of 15% < RH < 95%, we have adjusted a 4th order polynomial Eq. (6a) for adsorption and another 3rd order polynomial Eq. (6b) for desorption, using the Solver Excel tool by the GRG (Generalized Reduced Gradient) non-linear method. The large number of significant figures is necessary to obtain a good estimation of the EMC.

To deal with factor 2 (cotton EMC at different RH cott values) the values for each EMC from the adsorption and desorption are close enough to use the average of both.

Similarly, we have adjusted a 4th order equation polynomial (7) using the same fitting method as above to data of EMC versus RH cott values from Xie et al. [21 Fig. 8a], which are shown in our Fig. 4a:

To use these Eqs. (6a, 6b and 7) we need the RH cott_i−1 value calculated by Eq. (1b).

To solve the differential equation of RH cott = function (WADS, WDES, EMC, t) according the Eqs. (6a, 6b and 7) we use a numerical FEM method with the same time steps used at “Humidity change at the void from the vapour permeation through the canvas (Partial process 1)” and “Humidity change at the void from the air infiltration flow (Partial process 2)” sections and as explained in “Calculation procedure by time steps” section. The calculations in this case are more complex as the variable EMC implies a set of 4 conditionals for each of the three secondary variables: the moisture content of the cotton layer (MCCott%i), its change (ΔSCott%_i), and the change of moisture of the void (ΔSVCott%_i), listed in Table 1. These conditionals ensure, firstly, that MC% = < EMC% in each time step. Secondly, that if MC% > EMC% the remaining vapour goes to the air void, as it is not adsorbed by the cotton. The equations used to implement these conditionals within the Excel model tool can be seen in Additional file 1. Note than, in order to obtain the sorption and desorption rates with the correct units, the rate in (% rate/min) needs to be multiplied by Δts ∙ 60.

From the ΔSVoidCott%_i value, we find the vapour mass variation at void from cotton sorption in step “i” (ΔSVoid Cott_i) as:

After we calculate the Mvap cott_i−1 (void vapour mass from cotton sorption before the “i” step):

Humidity change at the void from the vapour sorption in the linen canvas (Partial process 4)

The linen canvas vapour sorption is the fourth partial process involved. Based on the same process described for cotton vapour sorption, we have adjusted the equations for linen vapour sorption rate (%/min) versus RH linen and for EMC versus RH linen. The polynomial equations are adjusted from the values at Xie et al. [21 Figs. 4c and 8c].

The equations for linen adsorption WADS (%rate/min) and desorption WDES (%rate/min) are:

For linen, the EMC versus RH canv graphic is shown in our Fig. 4b and the equation obtained is:

To use these Eqs. (10a, 10b and 11) we need the RH canv_i−1 calculated by Eq. (1b). As above, the differential equation for RH canv = function (WADS, WDES, EMC, t) according the Eqs. (10a, 10b and 11) is solved with a numerical FEM method. In this process the variable EMC, also implies a set of 4 conditionals for each of the three secondary variables (SorpCanv_i%, MCCanv_i% and ΔSVCanv%_i). These variables are calculated with the same conditionals expressed in Table 1.

From the ΔSVoidCanv%_i value, we find the vapour mass variation at void from canvas sorption in step “i” (ΔSVoid Canv_i) as:

Following this, we calculate the Mvap canv_i−1, which is the void vapour mass from canvas sorption before the “i” step, as well as the increase in this time step:

The general concepts explained in points 3.5 and 3.6, related to the sorption process (adsorption and desorption) and EMC in cellulose fibers are applicable to others fibers with different hygroscopic properties, which will require updated equations for adsorption–desorption. Similar data can be found in Xie et al. [21, Figs. 4 and 8] who calculated the sorption rate and EMC values for other fibres such as as hemp or jute.

Total MR at void and RH at the different layers at the end of each time step

The differential equations modelling the mixing ratio MR changes along the temperature cycles have been solved numerically for each of the partial processes equations in parallel as shown from “Humidity change at the void from the vapour permeation through the canvas (Partial process 1)” to “Humidity change at the void from the vapour sorption in the linen canvas (Partial process 4)” sections. The total MR change at the void is the sum of the changes caused by all the partial processes, which leads to the following expression for Cases 1 and 3 as:

This equation is slightly modified for Case 2 and 4, by replacing the increase in moisture through sorption in the canvas (ΔMR void sorp canv) by the same increase in the cotton layer (ΔMR void sorp cott). The rationale for this change is described in “Cycle temperatures calculation for each time step” section.

Knowing the MR allows the calculation of the Relative Humidity at the air boundary layer adjacent to the cotton layer, the canvas and aluminium (RH cott_i, RH canv_i and RH alum_i) at the end of the “i” time step and at the start of time step “i + 1”. From MR void_i we calculate the RH of each layer (RH layer_i) with Eq. (1b) for the canvas, cotton, or aluminium as layers, with the added conditionals:

This completes the calculations within the current time step “i”, for which we have obtained MR void_i, RH cott_i RH canv_i and RH alum_i. With the value of MR void in time step “i”, and the values of T room, T canv, T cott, T alum and MR room in the next time step “i + 1”, we can calculate first the new RH values at room, canvas, cotton and aluminium with Eq. (1b) and after all the variable values of the time step “i + 1” with the same equations as in time step “i”. We can repeat the same calculation process over one or more cycles.

In order to clarify the complete model process calculations, we include in Fig. 5 a diagram with the calculation variables and the reference of the equations involved in time step “i”.

Software used for the simulation model

The simulation can be done with software like Visual Basic, Excel macros, Python, or others. We have used Excel files, one for each case, carrying out each time step calculations for Vapour Canvas permeation + Air infiltration + Cotton sorption + Canvas sorption and the final step MR void in one line. We have chosen Excel because it is ubiquitous, user-friendly, and accessible. In order to obtain the required accuracy and a stable calculation along the time it is necessary to adjust the value of the time-step, Δt, which is different in each studied Case.

• Case 1, 2 (2 cycles of 4 h each): Δt = 0.00033 h, Δts = 1.2 s (48,000 Excel lines).

• Case 3 with 2 cycles of 24 h each: Δt = 0.001 h, Δts = 3.6 s (48,000 Excel lines).

• Case 4 (2 cycles of 24 h each) at Δt = 0.00025 h, Δts = 0.9 (192,000 Excel lines).

This number of rows is supported by the Excel tool. Longer experiments could potentially reach the limit of rows, and thus require alternative software.

Model parameters

To check our simulation model, we have used in all the cases the same dimensions and materials as in Padfield [16], a commercial canvas (0.6 m × 0.5 m) with a factory applied ground and a white acrylic paint layer, weighting 134 g and 0.447 kg/m². As this commercial canvas has one open woven layer of linen and other layer of ground, we have assumed in our calculations a density of 0.335 kg/m² for the linen canvas which corresponds to a 75% of the total. Thus, we estimate that the mass of canvas available for sorption and permeation is 100.5 g, while the mass of the ground layer is 33.5 g.

For vapour permeance flow, the white acrylic paint layer is the main barrier to the vapour flux compared with the other layers as the canvas and the factory applied ground (see Table 2). According to it, the vapour permeance used is 9.90E−11 kg/(m² s Pa) from the acrylic primed canvas at [12] and correspond to the material used at [16]. The void depth between the canvas and the aluminium back plate is 29 mm and the real void volume (excluding the stretcher and the cross-beam) is 0.00669 m³.

Our model can deal with permeance values from 1.00−E13 to 1.00E−09 kg/(m² s Pa) which are normal in the treated canvas used by painters. Around 1.00E−08 may appear instabilities in the calculations.

In our simulation models corresponding to [16 Figs.7, 11, 14 and 15] we use a 0.4 Vol/h air infiltration rate that corresponds to a value obtained at [16] by flushing nitrogen into the void and measuring the oxygen return by an optical method.

The woven cotton layer (0.632 kg/m² and 189 g) used in our models is located inside the void and stuck to the aluminium back plate in Case 2, and at 3 mm gap of the canvas in Case 4, as in [16 figs. 11 and 15] corresponding to ours Figs. 7d and 9d.