1)For the forced convection mass transfer example given in class, begin with the simplified...

Question

1)For the forced convection mass transfer example given in class, begin with the simplified governing equation derived in class along with the boundary conditions, and derive the solution given for this problemThe problem can be approximatedas:??????????????????=????????????????2Boundary conditions:Atz=0,????=0At x=0,????=??????At ???8,????=0This problem becomes a semi-infinite problem thatcan be solvedwith the similarity variable.The solution is:??????????=1-??????[??v4??????????????]Or it can be:??????????=????????[??v4??????????????]Which is the complimentary function.Now please show step by step how to get the above solutions.
	
Document Preview: 1) For the forced convection mass transfer example given in class, begin with the simplified governing 
equation derived in class along with the boundary conditions, and derive the solution given for this 
problem 
 
The problem can be approximated as: 
?? ?? ?? ??
?? ??
?? = ?? 
??????
2
?? ?? ?? ??
 
Boundary conditions: 
At       z=0,           ?? =0 
??
At       x=0,            ?? =?? 
?? ?? ??
At ?? ? 8,            ?? = 0        
??
 
????This problem becomes a semi-infinite problem that can be solved with the similarity variable. 
The solution is: 
  
?? ??
??
  
= 1 - ??????  
??
4??
?? ??  
?? ??
v
??
[ ]
??????
Or  it can be:

?? ??
??
  
= ?????? ??  
??
?? ?? 4?? 
?? ??
v
[ ?? ]
??????
Which  is the complimentary function. 
Now please show step by step how to get the above solutions.

Robert · Accepted Answer

Problem 1 
????????  
??????
????
 = ??????  
??2????
????2
 
Solution 
Assumptions to be used in a semi infinite Problem 
 constant thermophysical properties 
 no internal heat generation 
 uniform thermal conditions on its exposed surface 
 initially a uniform Concentration factor of CA throughout. 
As the given problem is a semi-infinite problem hence we need to consider a similarity variable to solve this 
problem 
Equation to be solved is  
 
????????  
??????
????
 = ??????  
??2????
????2
   
 
??2????
????2
 = ????????
??????  
 ??????
????
 
 
On comparing above equation with standard equation 
  
????????
??????
 = ??
?? 
 ??????
????
   ---------------------------(1) 
 
We get 
 
?? = ??????
???????? 
 
 
Now the Given boundry conditions are 
 
Mass transfer in this case occurs only in the direction normal to the surface (the x direction) Hence it will be 
considered as one-dimensional problem. 
So the  partial differential governing equation can be converted into an ordinary differential equation by 
combining the two independent variables x and z into a single variable η,

1)For the forced convection mass transfer example given in class, begin with the simplified governing equation derived in class along with the boundary conditions, and derive the solution given for...

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