Method of calculating the dimensions of greenhouse-type single slope watermaker by taking into account the accumulation of solar energy parnik tipidagi bir nishabli suv chuchutgichi o

Statement of the problem

Consider the static problem of thermoelasticity for isotropic bodies. It consists of the equilibrium equations [2] ³ ^ij
j1 _xi  X _i  0, i 1,3 (1)
the constitutive relation of Dughamel-Neumann representing the relationship between stresses and strains taking into account temperature
    _ij  _ij 2 _ij 3 2  (T T₀)_ij (2)
Cauchy relations

1. 
   u_i u _j
   

_ij  (  ) (3)

2. 
   xj xi
   

and boundary conditions

3. 
   u_{i 1}  u_i^o , _ijn_{j }2  S_i^o (4) ^j1
   

where, corresponds to thermal expansion coefficient, _ij stress tensor,_ij strain tensor, u_i displacement, T_temperature, T₀ initial temperature X_i volume force, , Lame constants, spherical part of strain tensor, n_j - outward normal to surface ₂ , S S S₁, ₂ , ₃ - external load vector components, _ij  delta Kronecker symbol and     (3 2 ).
The boundary value problem of thermoelasticity (1–4) can be reduced to a system of three
differential equations for displacements, i.e.
 ²u ²u ²u   ²v ²w  T

(2)x₂ y2  z2 ()xy  xz ^x  0
 
^ ²v ²v  ²v  ²u  ²w T

x2  z2 (2) y2 ()xy  yz y  0 (5)

^^x²w2  y²w2 ()x²uz  y²vz (2)z²w2 ^Tz  0.
 
Differential equations (5) can be considered with boundary conditions in displacements, stresses or mixed type. In our case, we consider a parallelepiped with a free boundary from loads, and the boundary conditions take the following form
 u v w 
11 x0,l1  (  2 ) x   y  z  (T T0) x0,l1  0

_{12x o l ,}1 ^ ^^uy  ^_x^{v }_ x o l ,1  0  (6)
13x o l_ ,1  u w 0 
^ z ^ _x ^_ x o l ,1  

_{ }

22 y0,l2  ux (  2 ) yv  wz  (T T0) y0,l2  0
 

 v u  
_{21y o l ,2} ^ x  _y _ y o l ,2  0  (7)
 v w 
23y o l_ ,2 ^ z ^ _y ^_ y o l ,2  0  _{ }
33 z0,l3   ux  yv (  2 ) wz (T T0) z0,l3  0
 
_{31z o l ,3} ^ ^^wx ^{ }^uz ^_ z o l ,3  0 . (8)
 w v  
32z o l_ ,3 ^ y ^ z ^ z o l ,3  0 




Fig. 1.
where l_i , i 1,3–are the lengths of the edges of the considered parallelepiped.

Numerical implementation

Replacing the derivatives in the boundary value problem (5–8) with difference relations, one can find the basic difference equations [3]
ui1, ,j k 2ui j k, , ui1, ,j k  ui j, 1, k 2ui j k, , ui j, 1, k ui j k, , 1 2ui j k, , ui j k, , 1 
( 2 )

  ₂  h₁ 
 vi 1, 1,j k vi 1, 1,j k vi 1, 1,j k vi 1, 1,j k

(  )   4hh_{1 2}
Ti1, ,j k Ti1, ,j k
  0
2h₁
2  ₂ 
h2 h3 
wi1, ,j k1  wi1, ,j k1  wi1, ,j k1  wi1, ,j k1 
 (9)
4h1 3h 
vi j k, , 1 2vi j k, , vi j k, , 1 

 ₂  h₃ 
 w  w 

vi j, 1, k 2vi j k, , vi j, 1, k
( 2 )

 vi1, ,j k 2vi j k, , vi1, ,j k

  2  2 h2  h1
 u u u u w  w

 ^  i 1, 1,j k i 1, 1,j k i 1, 1,j k i 1, 1,j k  4hh_{1 2}
Ti j, 1, k Ti j, 1, k
  0
2h₂
 i j, 1, 1 k i j, 1, 1 k i j, 1, 1 k i j, 1, 1 k _ (10)
4h h_{2 3} 

wi j k, , 1 2wi j k, ,  wi j k, , 1  wi1, ,j k 2wi j k, ,  wi1, ,j k wi j, 1, k 2wi j k, ,  wi j, 1, k 

( _ 2 ) ₂ ^ ₂  ₂  h3  h1 h2 
 ui1, ,j k1 ui1, ,j k1 ui1, ,j k1 ui1, ,j k1 vi j, 1, 1 k vi j, 1, 1 k vi j, 1, 1 k vi j, 1, 1 k 
^ ^    (11)
 4hh_{1 3} 4hh_{1 2} 
Ti j k, , 1 Ti j k, , 1
  0
2h₃
boundary conditions (6)

(_{ } 2 ) u N( 1, , )j k u N( ₁ 1, ,j k) _v N( ₁, j 1,k)v N( ₁, j 1,k) h1 2h
2 
w N( ¹, ,j k 1) w N( ¹, ,j k 1) 
 (T_N1, _j T₀ )  0 
2h₃

 u N( ₁, j 1,k)u N( ₁, j 1,k) v N( ₁, , )j k v N( ₁ 1, ,j k)   (12)
^    0
 2h₂ h₁  
  u N( ₁, ,j k 1)u N( ₁, ,j k 1) w N( ₁, , )j k w N( ₁ 1, ,j k)  

^    0   2h₃ h₁  
The finite-difference equations for the boundary conditions (7) and (8) can be written in a similar way.
Solving equations (9–12) with respect to u_{i j k, ,} , v_{i j k, ,} , w_{i j k, ,} we can construct the following iterative process

(n1)  ui(n1), ,j k ui(n1), ,j k  ui j(,n)1,k ui j(,n)1,k ui j k(, ,n) 1 ui j k(, ,n) 1   ui j k, , (  2 ) 2  2  2  

 h₁  h₂ h₃  

    
  4hh1 2 4h h2 3 Ti j, 1,k Ti j, 1,k  2(    2 ) 2 2   2h  /  h2  h2  h2  2   2 1 3   w  w  w  w w  w  (n) (n) (n) (n) (n) (n) (n1) i j k, , 1 i j k, , 1 i1, ,j k i1, ,j k i j, 1,k i j, 1,k wi j k, ,  (  2 ) 2  2  2   h3  h1 h2   u(n) u(n) u(n) u(n) v(n) v(n) v(n) v(n)	 
	 	        

    
  4hh1 2 T T  2(    2 ) 2 2   2h  /  h2  h2  h2  i1, ,j k i1, ,j k 1   1 2 3   v(n) v(n)  v(n) v(n) (n1) i j, 1,k i j, 1,k i1, ,j k i1, ,j k vi j k, ,  (  2 ) 2  2  h2  h1 (n) (n) (n) (n) (n)	4hh1 3 (n) (n) v v   i j k, , 1 2 i j k, , 1  h3  (n) (n)	(n)	
				        
 u u u u w  w  w  w  

( ) vi( n1), j 1,k vi( n1), j 1,k vi( n1), j 1,k vi( n1), j 1,k wi(n1), ,j k1  wi(n1), ,j k1  wi(n1), ,j k1  wi(n1), ,j k1  

i 1, j 1,k i 1, j 1,k i 1, j 1,k i 1, j 1,k i j,  1,k 1 i j,  1,k 1 i j,  1,k 1 i j,  1,k 1

^ ^  i1, ,j k1 i1, ,j k1 i1, ,j k1 i1, ,j k1  i j,  1,k 1 i j,  1,k 1 i j, 1,k1 i j,  1,k 1  
_ 4hh_{1 3} 4hh_{1 2}   (13)

Ti j k, , 12hTi j k, , 1  / 2(   2 2 )  2 2  2 2  
   h3 h1 h3  

u (N , , )j k  u (N 1, ,j k)
1 1   2  2h2 w^(^n^)(N1, ,j k 1) w^(^n^)(N1, ,j k 1)   (TN1, j T0 ) 2h3  h v^(^n^^1)(N1, , )j k  v^(^n^)(N1 1, ,j k) ^1u^(^n^)(N1, j 1,k)u^(^n^)(N1, j 1,k) 2h2	      (14)   

3   h  v(n) (N , j 1,k)v(n) (N , j 1,k)
(n1) (n) 1 1 1

(n1) (n) h1 (n) (n)  w (N1, , )j k  w (N1 1, ,j k) 2h₃ u (N1, ,j k 1)u (N1, ,j k 1) 
Equations (14) are written for a face of a parallelepiped at x  l₁ . For the rest of the faces, the
corresponding relations can be found in a similar way.