The inner surface of a combustion chamber wall receives heat from the products of combustion. The wall is being cooled by a coolant on the outer side. Compute the overall heat tran... The inner surface of a combustion chamber wall receives heat from the products of combustion. The wall is being cooled by a coolant on the outer side. Compute the overall heat transfer coefficient and draw the equivalent thermal circuit.
Understand the Problem
The question is asking for the calculation of the overall heat transfer coefficient for a combustion chamber wall that receives heat from combustion products while being cooled by a coolant on the outer side. Additionally, it requires the drawing of the equivalent thermal circuit to visualize the heat transfer processes involved.
Answer
The overall heat transfer coefficient is given by: $$ U = \frac{1}{\frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}} $$
Answer for screen readers
The overall heat transfer coefficient is given by:
$$ U = \frac{1}{\frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}} $$
Steps to Solve
- Identify the necessary parameters
To calculate the overall heat transfer coefficient, we will need to define the parameters:
- Heat transfer coefficient on the inner side ($h_i$) from combustion products.
- Heat transfer coefficient on the outer side ($h_o$) from the coolant.
- Thermal conductivity ($k$) of the combustion chamber wall material.
- Thickness of the wall ($L$).
- Set up the resistance model
The overall heat transfer can be represented using thermal resistances. The thermal resistances in series are:
- Inner convective resistance:
$$ R_i = \frac{1}{h_i A} $$
- Conduction resistance through the wall:
$$ R_{cond} = \frac{L}{k A} $$
- Outer convective resistance:
$$ R_o = \frac{1}{h_o A} $$
- Calculate the total thermal resistance
The total thermal resistance ($R_{total}$) is the sum of all resistances:
$$ R_{total} = R_i + R_{cond} + R_o $$
Substituting the formulas gives:
$$ R_{total} = \frac{1}{h_i A} + \frac{L}{k A} + \frac{1}{h_o A} $$
- Calculate the overall heat transfer coefficient
The overall heat transfer coefficient ($U$) can be calculated from total resistance as follows:
$$ U = \frac{1}{R_{total}} $$
- Simplify the equation
Substituting the expression for $R_{total}$:
$$ U = \frac{1}{\frac{1}{h_i A} + \frac{L}{k A} + \frac{1}{h_o A}} $$
This can be simplified to:
$$ U = \frac{1}{\frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}} $$
- Draw the equivalent thermal circuit
The equivalent thermal circuit can be represented with resistances.
- Draw three consecutive blocks, each representing:
- Convective resistance $R_i$ on the inner surface
- Conductive resistance $R_{cond}$ through the wall
- Convective resistance $R_o$ on the outer surface
Each block will connect from the inner surface temperature to the outer surface temperature, illustrating the flow of heat.
The overall heat transfer coefficient is given by:
$$ U = \frac{1}{\frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}} $$
More Information
The overall heat transfer coefficient quantifies the heat transfer effectiveness through the combustion chamber wall. Higher values of $U$ indicate better heat transfer.
Tips
- Not including all resistances in series.
- Forgetting to convert units, especially when dealing with conductivities and thickness.
- Misapplying the formula for resistance, particularly in cases of combined conduction and convection.
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