Chemical-plant equipment costs vary to size

In summary, the answer to the first question is that you would get the total cost of the shell multiplied by the density of the shell metal. The answer to the second question is that you would get the total cost of the shell multiplied by the density of the shell metal.
  • #1
Ortanul
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Homework Statement


Chemical-plant equipment costs rarely vary in proportion to (i.e., linearly with) size. In the

simplest case, cost varies with size according to the allometric equation, C = aSb

The exponent is typically between 0 and 1. For a wide variety of equipment types, it is approximately 0.6.

  1. (a) For 0 < b < 1, show that cost per unit size decreases with increasing size, resulting in an economy of scale.
  2. (b) Consider the case of a spherical storage tank. The size is commonly measured by internal volume Vt. Show that b = 2/3. On what parameters or properties would you expect the quantity a to depend?

Homework Equations


formulas for derivative

The Attempt at a Solution


For part a, I calculated the derivative of C/S with respect to the size, which equals to (b-1)*a*Sb-2. As the value of b-1 is negative, the value of this derivative is always negative, and so the increase in size would result in a decrease in the cost per unit size.
I don't understand what part b is asking about though. Compared with part a, it doesn't give anything apart from telling me that the tank is spherical. I have completely no clue about how to obtain a specific value for b without any additional information.
Any help would be much appreciated!
 
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  • #2
@Ortanul, Hi

What is the volume of a sphere of radius r?
Then, what other equation of a sphere could one apply that might indicate its cost with regard to its cost of material. How would you find that?
 
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  • #3
@265bits
Thanks for your reply!
The equation for calculating the volume is 4/3*pi*r^3
While for the other equation... Could you please give me a hint?
 
  • #4
256bits said:
@Ortanul, Hi

What is the volume of a sphere of radius r?
Then, what other equation of a sphere could one apply that might indicate its cost with regard to its cost of material. How would you find that?

Thanks for your reply!
The equation for calculating the volume is 4/3*pi*r^3
While for the other equation... Could you please give me a hint?
 
  • #5
Ortanul said:
Thanks for your reply!
The equation for calculating the volume is 4/3*pi*r^3
While for the other equation... Could you please give me a hint?
What is the volume of the shell. What do you get if you multiply this by the density of the shell metal?
 
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Related to Chemical-plant equipment costs vary to size

What factors affect the cost of chemical-plant equipment?

The cost of chemical-plant equipment can vary greatly depending on factors such as the size and complexity of the equipment, the materials used, the location of the plant, and the current market conditions.

How does the size of the equipment impact its cost?

The size of the equipment is a major determinant of its cost. Generally, larger equipment will require more materials and labor, resulting in a higher cost. Additionally, larger equipment may also require more specialized components and construction techniques, further driving up the overall cost.

Why do materials play a significant role in equipment costs?

The materials used in chemical-plant equipment can greatly impact its cost. High-quality and durable materials, such as stainless steel, can be more expensive but are necessary for the equipment to withstand harsh chemical environments and maintain its integrity over time.

How does the location of the plant affect equipment costs?

The location of the plant can also influence equipment costs. For example, a plant located in a remote or difficult-to-access area may incur higher transportation costs for the equipment and materials. Additionally, the availability of local resources and labor can also impact the cost of equipment.

What role do market conditions play in equipment costs?

Market conditions, such as supply and demand, can also affect the cost of chemical-plant equipment. During times of high demand, the cost of materials and labor may increase, resulting in higher equipment costs. Conversely, during a downturn in the market, equipment costs may decrease as suppliers lower their prices to remain competitive.

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