# Fundamentals of Momentum, Heat and Mass Transfer, 6th Edition International Student Version (6th Edition) Edit edition 93% (74 ratings) for this chapter’s solutionsSolutions for Chapter 19

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Chapter: Problem:

An immersion heater, rated at 1000 W, is in the form of a rectangular solid with dimensions 16 cm by 10 cm by 1 cm. Determine the surface temperature of the heater if it is oriented in 295 K water with

a. the 16-cm dimension vertical

b. the 10-cm dimension vertical

Step-by-step solution:
Chapter: Problem:
• Step 1 of 17

As the width of the solid is very less compared to the length and height neglect the heat transmitted by the ends of the solid.

Convert the height of the solid to meter:

The height of the solid in meters is 0.16 m.

Convert the length of the solid to meter:

The height of the solid in meters is 0.1 m.
• Step 2 of 17

Calculate the area of the rectangular solid, A from which heat transmission occurs by the formula:

Here, L is length of the solid and H is the height of the solid.

Substitute 0.1 m for L and 0.16 m for H in the above equation.

The area of the rectangular solid is
• Step 3 of 17

Calculate the heat flux, by substituting 1000 W for q and for A:

• Step 4 of 17
Therefore, the heat flux is .
• Step 5 of 17

As the value of surface temperature is unknown we need to solve the problem by the use trial and error technique by assuming a value for surface temperature for property evaluation and then solving for T .Therefore assume .

Find the film temperature, on which the properties of water will be calculated by the formula:

Here is the temperature of the surface of the heater and is the temperature of the surrounding water

Substitute 330 K for and 295 K for in the above equation

• Step 6 of 17
Therefore, the film temperature is 313 K.
• Step 7 of 17

a) Calculate the surface temperature of the rectangular heater when its 16 cm dimension is vertical as follows:

Calculate the Grashof number, from the formula:

…… (1)

Here, is the coefficient of thermal expansion, g is the gravitational acceleration, is the kinematic viscosity of water, L is the characteristic length and is the temperature difference between the surface temperature and the surrounding water temperature .

• Step 8 of 17

Look for the value of in the table of physical properties of gases and liquids for water at the temperature of 313 K

For , the value of is .

The characteristic length in this case is the vertical dimension, .

Therefore,

Find the temperature difference, by the formula:

Here is the temperature of the surface of the heater and is the temperature of the surrounding water

Substitute 330 K for and 295 K for in the above equation

Therefore, the temperature difference is 35 K.

• Step 9 of 17

Substitute for , 0.16 m for L and 35 K for in equation (1)

Therefore, the Grashof number is .

• Step 10 of 17

Calculate the Rayleigh number, from the formula:

…… (2)

Here, is the Grashof number and Pr is the Prandtl number

Look for the value of Pr in the table of physical properties of gases and liquids for water at the temperature of 313 K

For , the value of Pr is 4.33.

Substitute for and 4.33 for Pr in equation (2),

Therefore, the Rayleigh number is .

• Step 11 of 17

Find the Nusselt number for a rectangular surface in vertical orientation by using the equation,

Substitute for and 4.33 for Pr in the above equation,

Therefore, the Nusselt number is 251.31.

• Step 12 of 17

The heat transfer coefficient, h for a rectangular surface in vertical orientation is

…… (3)

Here, k is the thermal conductivity of water.

Look for the value of k in the table of physical properties of gases and liquids for water at the temperature of 313 K

For , the value of k is .

Substitute for k, 0.16 m for L and 251.31 for in the equation (3).

Therefore, the heat transfer Coefficient is .

Now, express the convective heat flux,

Rearrange the above equation to obtain ,

…… (4)

• Step 13 of 17

Substitute for , 295 K for and for h in equation (4),

Thus, the surface temperature of the rectangular solid heater when its 16 cm dimension is vertical is .

• Step 14 of 17

b) Calculate the surface temperature of the rectangular solid heater when its 10 cm dimension is oriented vertically as follows:

The characteristic length in this case is the vertical dimension, .

Therefore,

Substitute for , 0.1 m for L, 35 K for in equation (1), to calculate the Grashof number as follows:

Therefore, the Grashof number is .

• Step 15 of 17

Substitute for and 4.33 for Pr in equation (2), to calculate the Rayleigh number as follows:

Therefore, the Rayleigh number is .

• Step 16 of 17

The Nusselt number for a rectangular surface in vertical orientation is

Substitute for and 4.33 for Pr in the above equation,

Therefore, the Nusselt number is 161.64.

• Step 17 of 17

Substitute for k, 0.1 m for L and 161.64 for in the equation (3), to calculate the heat transfer coefficient as follows:

Therefore, the heat transfer Coefficient is.

Substitute for h, 295 K for and for in equation (4), to calculate the surface temperature as follows:

Thus, the surface temperature of the rectangular solid heater when its 10 cm dimension is vertical is .

Corresponding Textbook

Fundamentals of Momentum, Heat and Mass Transfer, 6th Edition International Student Version | 6th Edition
9781118808870ISBN-13: 1118808878ISBN: James WeltyAuthors:
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