The assignment is of Computational mechanics. Related to python. I need task 1-3 by hand and other ones in python.

The University of Queensland

School of Mechanical and Mining Engineering

MECH2700 Computational Engineering and Data Analysis (2021)

Assignment: Computational Heat Transfer

Introduction

The Queensland government is considering building 100 MW solar thermal power plants (Figure 1) with thermal storage that can continuously supply the majority of electricity to the State. Figure 2 shows the overall plant operating cycle. A field of sun tracking mirrors (heliostats) focuses incoming solar radiation on a receiver located atop a central tower, which absorbs the radiation and transfers the heat to a flow of molten salt. In order to produce power continually, the solar thermal plant stores heat in a large tank of hot molten salt at temperature TH = 820 K. Hot salt is then pumped from this tank to continuously supply heat to a steam turbine that generates power. After transferring heat to the turbine working fluid, the salt is cooled to temperature TL = 560 K and is pumped into the “cold” salt storage tank, where is awaits reheating in the receiver. In this assignment, you will conduct a heat transfer analysis that will allow you to size the receiver so that it can survive peak solar heating.

Figure 1: Crescent Dunes Solar Energy Project as seen from an airliner (Author: Amble. Source: en.wikipedia.org).

The thermal conditions within the receiver are extreme. Even using modern high-temperature materials to construct the receiver (carbon-carbon composite in this case), it is a challenge to prevent the wall surface temperature exceeding the material limit (1800 K here). Safe operation can be achieved by two complementary approaches: rapidly transferring heat to the molten salt and spreading the incoming radiative heating Q? rad in over a suitably large receiver surface area

Figure 2: Operating cycle of solar thermal plant with molten salt thermal storage.

Arec. In this assignment you will analyse the effectiveness of fins for heat transfer enhancement and then determine the required Arec based on the outcome.

A fin-type element is one device used in engineering design to enhance the heat transfer rate when large amounts of heat require dissipating from a component. The extended surface area of a fin allows it to readily transfer heat to the surroundings (or absorb, depending on the application). A typical use for fins is on the outer surface of an engine. For conventional engines such as those found in motorcycles and certain propeller-driven aircraft, the fins are air-cooled. For the solar thermal receiver considered here, the fins are cooled by molten salt.

In Figure 3, a fin of uniform cross-sectional area is shown. The length, L, is 20.0cm and the thickness, t, is 2.0cm. The cross-section is rectangular with a depth d = 10.0m (note that in reality the receiver is cylindrical, but the effects of curvature are negligible here).

.

Qrad in

---

Figure 3: A representation of fins protruding from a wall. Zoomed region shows pertinent dimensions for the one-dimensional problem.

You will design the receiver to survive the greatest incoming radiative heating at midday on a clear day: Q? rad in = I0Amirror, where I0 = 1000 W/m2 is the peak solar irradiation as sea level and Amirror = 6 × 105 m2 is the area of the mirror field. You will eventually size the receiver surface area such that for this peak heating, the wall temperature is Tw = 1600.0K (within a reasonable safety factor of the material limit). The heat from the receiver wall at Tw will be conducted along the fins and dissipated into the surrounding molten salt (acting as a coolant) which flows over the fins. You will consider the cooling of a fin where the molten salt temperature Ta has reached the hot salt storage temperature, Ta = TH = 820.0K. This corresponds to the the minimum cooling of the fin and therefore give us a conservative estimate of the heat that can be dissipated by each fin.

The aim of this assignment is to use a numerical solution procedure to estimate the temperature distribution in the fin, and from this compute the heat transfer rate provided by each fin. This can then be use to determine the surface area of the receiver over which Q? rad in must be spread.

The Mathematical Model

From the study of heat transfer, this problem may be recognised as a one-dimensional steadystate conduction problem. The equation for the distribution of temperature within the fin is:

. (1)

In Equation 1, h is the heat transfer coefficient taken as 15.0kW/m2K. The cross-sectional area is Ac = td, where the thickness is t and the depth is d. The perimeter, P = 2t + 2d, is the distance around cross-section. The thermal conductivity, k, may be assumed constant throughout the fin and is given as 250.0W/m.K, which is a representative value for carboncarbon composite in the plane of the fibers.

To complete the problem definition, the behaviour at the ends of the fins needs to be specified. The boundary conditions, in mathematical form, are:

(2)

(3)

The first condition states that the temperature at x = 0 is simply equal to the wall temperature of the receiver. The second condition comes from a balance of heat flux at the fin tip. The rate of heat conduction through the tip of the fin is equal to rate at which the heat is convectively absorbed by the molten salt at the tip surface.

The numerical solution method

The first step in the numerical solution method is to discretise the fin along its length into a number of equally spaced segments as shown in Figure 4. The joints of the segments will be called nodes and the spacing between nodes is ?x = L/n where n is the number of segments.

In the numerical solution procedure, we will compute the temperature at each of the solution nodes 1 through n. This is done by replacing the derivatives in the governing equation, Equation 1, and the boundary condition, Equation 3, by finite-difference approximations. Having formed finite-difference equations for each solution node, these equations may be assembled as a set of linear algebraic equations in standard matrix form AT~ =~b and solved for the vector of

Figure 4: Finite-difference nodes distributed along the fin. (unknown) temperatures

(4)

Assignment Tasks

Preliminary analysis

Task 1.

Show that the second derivative of T at node i may be approximated by the following finitedifference formula:

(5)

where Ti is shorthand for T(x = xi) and ?x is the node spacing.

The finite-difference form of the governing equation may now be written for all interior nodes, after substituion and rearrangement, as:

(6)

Note that the equations are linear in the unknown temperatures, Ti, and so may be assembled into matrix form. These equations are valid for all interior nodes, ie., up to i = n - 1. The final node, n, must be treated specially — the treatment of the end node will form Task 2.

Task 2.

Show that the first derivative of T may be approximated, using a one-sided difference, as:

(7)

Use this approximation in the boundary condition, Equation 3, to show that the finite-difference equation for the boundary node n is:

(8)

Task 3.

For n = 5, write out the five finite-difference equations (i = 1...5) and then assemble the matrix of coefficients, A, and the vector ~b in the system of equations, AT~ = ~b. Include a sixth line in your matrix system at the top: the trivial equation, T0 = Tw, for the node n = 0. This will be useful later when you try to build the matrix system in your computer code. What is special about the structure of matrix A?

Computer implementation

Task 4.

Write a Python program to solve for the temperatures in the fin using one of the direct solvers for systems of linear equations. Take advantage of the special structure of A to make your code more computationally efficient than the general method presented in lectures. In the discussion section of your report, comment on whether a direct solver was the most appropriate choice for this problem.

Task 5.

Plot the temperatures for n = 5,10,20,200 against the analytic solution:

where

(Source: The analytic solution may be found in the text by Incropera and De Witt, “Fundamentals of Heat and Mass Transfer”. Look for the section on “Fins of uniform cross-sectional area”. The fin in this problem is of finite length with a convective end boundary condition.)

Task 6.

As separate figures, plot the temperature at the fin tip and the relative error at the fin tip (kTn - Texact(L)k/Texact(L)) for a range of values of n to show convergence of the numerical solution as ?x = L/n approaches zero. What value of n is required to give a numerical solution that is within 0.1% of the analytic solution, in terms of the relative error in temperature at the convective boundary node?

Task 7.

Based on a heat flux balance, the heat transfer rate from the entire fin is equal to the heat transfer rate through the base of the fin (ie. the conduction from the receiver wall into the fin). Compute the heat transfer rate for the fin using the finite-difference approximation:

(9)

Note that T1 is the value at the first node computed as part of the solution procedure. Plot the value for heat transfer rate, Q? fin, as ?x approaches zero for the n values used in Task 5. What value of n is required to give a numerical estimate for heat transfer rate that is within 0.1% of the analytic solution:

.

If each fin dissipates heat from a section of receiver wall with area 5td, what must the total receiver wall area Arec be so that Q? rad in can be dissipated without the wall temperature exceeding our chosen limit. Comment on whether this surface area is reasonable for a plant of the scale under consideration.

Submission Notes and Assessment Criteria

Work individually or in groups of two for this assignment. Submit a single report for the group, detailing the problem, your analysis, code, results for Tasks 1 to 7 and a discussion of the approach and findings. This report should be brief but well structured, so that the tutor can easily interpret your results. The Discussion section should contain responses to the discussion questions asked in Tasks 1-7, along with a commentary on the accuracy and limitations of the modelling approach. Include your Python code and combine it all together into a single pdf file with a cover page that clearly tells us who you are. Scanned, hand-written documents with appropriate cut-and-paste items are perfectly acceptable, but plots should be computergenerated.

For assessment of your work, the tutors will be looking for answers to the following questions:

• Have you done what the tasks require? Note that your report should be well structuredand tidy so that the tutor can easily find your answers. Having your plots correctly titled and axes labeled also helps a lot here.

• Are your answers accurate?

• Is your code well-structured and clearly written, with suitable documentation comments?

• Does your code correspond to the results shown?

The University of Queensland

School of Mechanical and Mining Engineering

MECH2700 Computational Engineering and Data Analysis (2021)

Assignment: Computational Heat Transfer

Introduction

The Queensland government is considering building 100 MW solar thermal power plants (Figure 1) with thermal storage that can continuously supply the majority of electricity to the State. Figure 2 shows the overall plant operating cycle. A field of sun tracking mirrors (heliostats) focuses incoming solar radiation on a receiver located atop a central tower, which absorbs the radiation and transfers the heat to a flow of molten salt. In order to produce power continually, the solar thermal plant stores heat in a large tank of hot molten salt at temperature TH = 820 K. Hot salt is then pumped from this tank to continuously supply heat to a steam turbine that generates power. After transferring heat to the turbine working fluid, the salt is cooled to temperature TL = 560 K and is pumped into the “cold” salt storage tank, where is awaits reheating in the receiver. In this assignment, you will conduct a heat transfer analysis that will allow you to size the receiver so that it can survive peak solar heating.

Figure 1: Crescent Dunes Solar Energy Project as seen from an airliner (Author: Amble. Source: en.wikipedia.org).

The thermal conditions within the receiver are extreme. Even using modern high-temperature materials to construct the receiver (carbon-carbon composite in this case), it is a challenge to prevent the wall surface temperature exceeding the material limit (1800 K here). Safe operation can be achieved by two complementary approaches: rapidly transferring heat to the molten salt and spreading the incoming radiative heating Q? rad in over a suitably large receiver surface area

Figure 2: Operating cycle of solar thermal plant with molten salt thermal storage.

Arec. In this assignment you will analyse the effectiveness of fins for heat transfer enhancement and then determine the required Arec based on the outcome.

A fin-type element is one device used in engineering design to enhance the heat transfer rate when large amounts of heat require dissipating from a component. The extended surface area of a fin allows it to readily transfer heat to the surroundings (or absorb, depending on the application). A typical use for fins is on the outer surface of an engine. For conventional engines such as those found in motorcycles and certain propeller-driven aircraft, the fins are air-cooled. For the solar thermal receiver considered here, the fins are cooled by molten salt.

In Figure 3, a fin of uniform cross-sectional area is shown. The length, L, is 20.0cm and the thickness, t, is 2.0cm. The cross-section is rectangular with a depth d = 10.0m (note that in reality the receiver is cylindrical, but the effects of curvature are negligible here).

.

Qrad in

---

Figure 3: A representation of fins protruding from a wall. Zoomed region shows pertinent dimensions for the one-dimensional problem.

You will design the receiver to survive the greatest incoming radiative heating at midday on a clear day: Q? rad in = I0Amirror, where I0 = 1000 W/m2 is the peak solar irradiation as sea level and Amirror = 6 × 105 m2 is the area of the mirror field. You will eventually size the receiver surface area such that for this peak heating, the wall temperature is Tw = 1600.0K (within a reasonable safety factor of the material limit). The heat from the receiver wall at Tw will be conducted along the fins and dissipated into the surrounding molten salt (acting as a coolant) which flows over the fins. You will consider the cooling of a fin where the molten salt temperature Ta has reached the hot salt storage temperature, Ta = TH = 820.0K. This corresponds to the the minimum cooling of the fin and therefore give us a conservative estimate of the heat that can be dissipated by each fin.

The aim of this assignment is to use a numerical solution procedure to estimate the temperature distribution in the fin, and from this compute the heat transfer rate provided by each fin. This can then be use to determine the surface area of the receiver over which Q? rad in must be spread.

The Mathematical Model

From the study of heat transfer, this problem may be recognised as a one-dimensional steadystate conduction problem. The equation for the distribution of temperature within the fin is:

. (1)

In Equation 1, h is the heat transfer coefficient taken as 15.0kW/m2K. The cross-sectional area is Ac = td, where the thickness is t and the depth is d. The perimeter, P = 2t + 2d, is the distance around cross-section. The thermal conductivity, k, may be assumed constant throughout the fin and is given as 250.0W/m.K, which is a representative value for carboncarbon composite in the plane of the fibers.

To complete the problem definition, the behaviour at the ends of the fins needs to be specified. The boundary conditions, in mathematical form, are:

(2)

(3)

The first condition states that the temperature at x = 0 is simply equal to the wall temperature of the receiver. The second condition comes from a balance of heat flux at the fin tip. The rate of heat conduction through the tip of the fin is equal to rate at which the heat is convectively absorbed by the molten salt at the tip surface.

The numerical solution method

The first step in the numerical solution method is to discretise the fin along its length into a number of equally spaced segments as shown in Figure 4. The joints of the segments will be called nodes and the spacing between nodes is ?x = L/n where n is the number of segments.

In the numerical solution procedure, we will compute the temperature at each of the solution nodes 1 through n. This is done by replacing the derivatives in the governing equation, Equation 1, and the boundary condition, Equation 3, by finite-difference approximations. Having formed finite-difference equations for each solution node, these equations may be assembled as a set of linear algebraic equations in standard matrix form AT~ =~b and solved for the vector of

Figure 4: Finite-difference nodes distributed along the fin. (unknown) temperatures

(4)

Assignment Tasks

Preliminary analysis

Task 1.

Show that the second derivative of T at node i may be approximated by the following finitedifference formula:

(5)

where Ti is shorthand for T(x = xi) and ?x is the node spacing.

The finite-difference form of the governing equation may now be written for all interior nodes, after substituion and rearrangement, as:

(6)

Note that the equations are linear in the unknown temperatures, Ti, and so may be assembled into matrix form. These equations are valid for all interior nodes, ie., up to i = n - 1. The final node, n, must be treated specially — the treatment of the end node will form Task 2.

Task 2.

Show that the first derivative of T may be approximated, using a one-sided difference, as:

(7)

Use this approximation in the boundary condition, Equation 3, to show that the finite-difference equation for the boundary node n is:

(8)

Task 3.

For n = 5, write out the five finite-difference equations (i = 1...5) and then assemble the matrix of coefficients, A, and the vector ~b in the system of equations, AT~ = ~b. Include a sixth line in your matrix system at the top: the trivial equation, T0 = Tw, for the node n = 0. This will be useful later when you try to build the matrix system in your computer code. What is special about the structure of matrix A?

Computer implementation

Task 4.

Write a Python program to solve for the temperatures in the fin using one of the direct solvers for systems of linear equations. Take advantage of the special structure of A to make your code more computationally efficient than the general method presented in lectures. In the discussion section of your report, comment on whether a direct solver was the most appropriate choice for this problem.

Task 5.

Plot the temperatures for n = 5,10,20,200 against the analytic solution:

where

(Source: The analytic solution may be found in the text by Incropera and De Witt, “Fundamentals of Heat and Mass Transfer”. Look for the section on “Fins of uniform cross-sectional area”. The fin in this problem is of finite length with a convective end boundary condition.)

Task 6.

As separate figures, plot the temperature at the fin tip and the relative error at the fin tip (kTn - Texact(L)k/Texact(L)) for a range of values of n to show convergence of the numerical solution as ?x = L/n approaches zero. What value of n is required to give a numerical solution that is within 0.1% of the analytic solution, in terms of the relative error in temperature at the convective boundary node?

Task 7.

Based on a heat flux balance, the heat transfer rate from the entire fin is equal to the heat transfer rate through the base of the fin (ie. the conduction from the receiver wall into the fin). Compute the heat transfer rate for the fin using the finite-difference approximation:

(9)

Note that T1 is the value at the first node computed as part of the solution procedure. Plot the value for heat transfer rate, Q? fin, as ?x approaches zero for the n values used in Task 5. What value of n is required to give a numerical estimate for heat transfer rate that is within 0.1% of the analytic solution:

.

If each fin dissipates heat from a section of receiver wall with area 5td, what must the total receiver wall area Arec be so that Q? rad in can be dissipated without the wall temperature exceeding our chosen limit. Comment on whether this surface area is reasonable for a plant of the scale under consideration.

Submission Notes and Assessment Criteria

Work individually or in groups of two for this assignment. Submit a single report for the group, detailing the problem, your analysis, code, results for Tasks 1 to 7 and a discussion of the approach and findings. This report should be brief but well structured, so that the tutor can easily interpret your results. The Discussion section should contain responses to the discussion questions asked in Tasks 1-7, along with a commentary on the accuracy and limitations of the modelling approach. Include your Python code and combine it all together into a single pdf file with a cover page that clearly tells us who you are. Scanned, hand-written documents with appropriate cut-and-paste items are perfectly acceptable, but plots should be computergenerated.

For assessment of your work, the tutors will be looking for answers to the following questions:

• Have you done what the tasks require? Note that your report should be well structuredand tidy so that the tutor can easily find your answers. Having your plots correctly titled and axes labeled also helps a lot here.

• Are your answers accurate?

• Is your code well-structured and clearly written, with suitable documentation comments?

• Does your code correspond to the results shown?

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