Heat balance equations for cover, basin and drum

The following section represents the governing equations of the model. It includes all the heat balance equations for the cover, the basin and the rotating drum. The model of Malaeb et al (9) is adopted in this study, where modification were made to include surface roughness of the drum. The cover of the still is considered as one single part including the upper curved part and the two vertical sides, whereas the drum is discretized to get the temperature at each element of it.

The heat balance for the still cover, denoted as symbol c, can be written as in Equation (1):

m_c C_(p_c ) (dT_c)/dt=q_rad A_c-q_cd A_cd-q_cb A_cb-q_ca A_ca (1)

Where m_c refers to the mass of the cover, C_(p_c ) the heat capacity of the cover, T_c the cover temperature, q_rad A_c is the radiative heat transfer, A_c the cover area, q_cd the heat transfer between the cover and the drum (denoted as symbol d) and A_cd the area of the cover that interacts with the drum, q_cb the heat transfer between the brine water surface and the cover, A_cb the exposed area of the brine water surface i.

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e. excluding the brine diameter, q_ca the heat transfer between the cover and the ambient, A_ca the area of the cover exposed to the ambient.

The radiative heat transfer is defined as in Equation (2):

q_rad=?_c IA_c (2)

Where ?_c is the cover absorptivity and I is the solar radiation.

The heat transfer q_cd between the cover and each element i of the n drum elements of width dx and length L_d each is given by Equation (3):

q_cd=?_(i=1)^n??dxh_cd (T_di-T_ci ) L_d ? (3)

Where h_cd is the heat transfer coefficient between the cover and the drum.

The heat transfer qcb between the cover and the brine water surface (denoted as symbol b) that is of width Sb on each side of the drum is given by Equation (4):

qcb=h_cd (T_c-T_b ) A_cb=h_cd (T_c-T_b )(2S_b L) (4)

Where h_cd is the heat transfer coefficient between the cover and the water, T_b is the brine water temperature and L the length of the basin.

The heat transfer qga between the cover and ambient (denoted as symbol a) is given by Equation (5):

q_ca=h_ca (T_c-T_a ) A_g=h_ca (T_c-T_a )(?r_c+2S_c ) (5)

Where h_ca is the heat transfer between the cover and the ambient, T_a the ambient temperature, S_c the length of the vertical side of the cover and r_c the radius of the curved cover.

By substituting in the Equation (1):

[?_c (?r_c+2S_c ) ?tk?_c LC_(p_c ) (T_g2-T_g1)/dt]=?_c I_1 (?r_c L)+?_c I_2 (2S_c L)+h_bc (T_b-T_c2 )(2S_b L)-h_ba (T_c2-T_a )(?r_c+2S_c )L+?_(i=1)^n??dxh_cd (T_di-T_c1 ) L_d ? (6)

The heat transfer coefficient h_cais the summation of the radiation (h_rad )and convection terms between the still cover and the ambient, which are given by Equations (7) and (8) respectively:

h_rad=??(T_c^2+T_sky^2 )(T_c+T_sky ) (7)

h_ca=2.8+3v (8)

?=1/?_b +1/?_c -1 (9)

T_sky=T_a-10 (10)

where ? is the Stephan-Boltzmann constant, T_sky is the sky temperature, v is the wind speed, ?_b and ?_c are the emissivity of water and glass.

The heat transfer coefficient between the still cover and the water in the basin (hcb) is the summation of the radiation, evaporation and convection terms as following in Equation (11):

h_cb=h_(?rad?_cb+) h_(?evap?_cb+) h_(?conv?_cb ) (11)

The radiation term between water and cover is given by Equation (12):

h_(?rad?_cb )=??(T_c^2+T_b^2 )(T_c+T_b ) (12)

The evaporation term is given by Equation (13):

h_(?evap?_cb )=0.016273h_(?conv?_cb ) (p_c-p_b ) (13)

The convection term is given by Equation (14):

h_(?conv?_cb )=0.884[(T_c-T_b )+((p_c-p_b ) T_c)/(268.9??10?^3-p_c )]^(1/3) (14)

Where p_c and p_brepresent the partial vapor pressure at condensation and evaporation surface temperatures respectively.

The total heat transfer coefficient between the drum and cover (h_dc) is the summation of the individual coefficients representing radiation, evaporation and convection as in Equation (15):

h_dc=h_(?rad?_dc+) h_(?evap?_dc+) h_(?conv?_dc ) (15)

The radiation term between the drum element and the cover (h_(?rad?_dc ))is given by Equation (16):

h_(?rad?_dc )=??(T_c^2+T_di^2 )(T_c+T_di ) (16)

The evaporation term (h_(?evap?_dc )) is given by Equation (17):

h_(?evap?_dc )=0.016273h_(?conv?_cdi ) (p_c-p_cdi ) (17)

The convection term ?(h?_(?conv?_cb )) is given by Equation (18):

h_(?conv?_cb )=0.884[(T_c-T_di )+((p_c-p_di ) T_c)/(268.9??10?^3-p_c )]^(1/3) (18)

In addition to the above equations, the heat transfer q_(?conv?_dai ) has to be included in order to account for the convection between the drum surface and the air inside the hollow drum, which is at temperature T_ai.

This term is given by Equation (19):

h_(?conv?_dai )=0.884[(T_ai-T_di )+((p_ai-p_di ) T_ai)/(268.9??10?^3-p_ai )]^(1/3) (19)

The transient energy equation for a drum element then becomes as in Equation (20) and (21):

d/dt (m_d C_(p_d ) T+m_b C_(p_b ) T)=((q_(?cond?_x-) q_(?cond?_(x+?x) ) ))/?x+q_solar-q_(?conv?_dc )-q_(?rad?_dc )-q_(?evap?_dc )-q_(?conv?_dai ) (20)

Or:

[?_d t_(h_d ) LC_(p_d ) (T_d2-T_d1)/dt+?_b ?tk?_(b_1 ) LC_(p_b ) (T_d2-T_d1)/dt+?_b T_d1 LC_(p_b ) (?tk?_b2-?tk?_b1)/dt]=?I(t)-h_bc (T_b-T_c2 )(2S_b L)-k (T_d2-T_dbefore)/?tk?_d ?-k (T_d2-T_dafter)/?tk?_d -h?_d (T_d1-T_c )-h_(?cond?_dai ) (T_d1-T_ai ) (21)

For a drum element just coming into/ leaving the water in the basin, the heat balance equation is written as follows in Equation (22):

d/dt [m_d C_(p_d ) T]=h_bd (T_b-T_d )-k (T_d2-T_(d_before ))/?tk?_d -k (T_d2-T_(d_after ))/?tk?_d (22)

Where h_bd is the convection heat transfer coefficient between the drum element and the water or:

?_d t_(h_d ) C_(p_d ) (T_d2-T_(d_1 ))/dt=h_bd (T_d-T_b ) (23)

For the water in the basin, the heat balance equation can be written as in Equation (24):

d/dt [m_b C_(p_b ) T]=?_g IA_bc-q_bc A_bc-h_bd (T_b-T_d ) (24)

Regarding the radiative heat transfer between the surfaces, the view factor (F) should be taken in to consideration. It is defined as the fraction of radiation that is leaving surface i and intercepted by surface j. The involved surfaces in calculating their view factors are: cover (c); drum (d); water on one side of the system (b1); water on the other side of the system (b2); fiberglass on one vertical side of the system (f1) and fiberglass on the other vertical side of the system (f2). Using the summation rule, the view factors of the different parts of the system can be related as the following in Equations (25-30):

Glass Cover: Fcc + Fcd + Fcf1 + Fcf2 + Fcb1 + Fcb2 = 1 (25)

Drum: Fdc + Fdd + Fdf1 + Fdf2 + Fdb1 + Fdb2 = 1 (26)

Fiberglass side 1: Ff1c + Ff1d + Ff1f1 + Ff1f2 + Ff1b1 + Ff1b2 = 1 (27)

Fiberglass side 2: Ff2c + Ff2d + Ff2f1 + Ff2f2 + Ff2b1 + Ff2b2 = 1 (28)

Water surface 1: Fb1c + Fb1d + Fb1f1 + Fb1f2 + Fb1b1 + Fb1b2 = 1 (29)

Water surface 2: Fb2c+ Fb2d + Fb2f1 + Fb2f2 + Fb2b1 + Fb2b2 =1 (30)

By inspection: Fdd = Ff1f1 = Ff1f2 = Ff2f1 = Ff2f2 = Fb1b1 = Fb1b2 = Fb2b1 = Fb2b2 = 0.

In order to obtain the view factor between the water and the still cover, the total view factor (Fbc_total) of one side of the system is obtained. The still cover is divided into 2 parts: cover 1, which is the curved part of the cover and cover 2, which is the vertical side of the cover.

The total view factor Fbc_total is the sum of view factors for the water-system side and water-glass1 (g1): Fbf_total =Fbf + Fbg2.

Fbf is calculated from Figure 3 whereby X = 1.5, Y = 0.035 and Zbf = 0.12, which gives: Fbf = 0.45.

Figure 3: View Factor for Perpendicular Rectangles (30)

Similarly, Fbf_total = 0.48 is found using Zbf_total = 0.42. Therefore, Fbg2 = 0.03.

Figure 4 is used to find Fbg1, whereby glass 1 is approximated as a horizontal plane parallel to the water surface plane. For this case, X = 0.035, Y = 1.5 and L = 0.47.

Figure 4: View Factor for Aligned Parallel Rectangles (30)

In order to calculate the new film thickness formed around the rough corrugated rotating drum, the work of Hay et al. (22) will be integrated within the model. However, the equations will have to be twisted in order to be able to apply them in our model as represented in the following section.