ABEE3037 – 19/20 CFD for the Built Environment Dr James Pinchin Email: [email protected] Page 1 of 4 COURSEWORK 1 – HEAT CALCULATIONS Key Dates Briefing – 24/10/19 Submission – Midday on 7/11/2019 Feedback – In the lecture on 21/11/19. Marking & Submission A marking rubric can be found at the end of this document. Submission should done be via the course Moodle page. A report giving answers to the questions below should be submitted in word (.doc) or PDF (.pdf) format. You may present hand written answers which have been scanned to PDF format. The first lines of the report should give the title: “ABEE3037 2019/20 – Coursework 1”, then your name, student number and condition assignment (letter). Setup In the first part of this coursework we’ll be using two geometries, both are walls with a total thickness of 0.3 m. In the first wall (1) the construction material is thermally homogeneous with a thermal diffusivity (α) equal to the value given to you in the condition assignment which can be found on Moodle. In the second wall (2) the same material is used for the two 10cm thick outer layers but the layers are separated by a 10cm thick air gap with α = 1.9×10-5 ms-2. (1) (2) In both cases the temperatures are known on either side of the wall. On the left side (x = 0) the temperature is 10oC, on the right edge (x=L) the temperature is 30oC. ABEE3037 – 19/20 CFD for the Built Environment Dr James Pinchin Email: [email protected] Page 2 of 4 Heat Equation – Steady State In this section you will form and solve a system of equations for the temperature across the wall when in steady state. 1) a. Write down the 1D steady state heat equation after discretisation using the finite difference method, central difference approach. Write down a set of simultaneous equations for the first wall (1) using five nodes with equal spacing. b. Express the simultaneous equations in the matrix form, AT = b. c. Using values from the walls specified above, solve this system of equations using the direct approach. You may use Matlab or any other tool to perform the inversion of A and multiplication of b. Give the temperature at each node in degrees centigrade to two decimal places. d. Write down the 1D steady state heat equation after discretisation using the finite difference method, central difference approach. Write down a set of simultaneous equations for the second wall (2). Choose the number of nodes to use and their spacing (you should need no more than 10 nodes). e. Justify your choice of node placement / number of nodes. f. Express the simultaneous equations in the matrix form, AT = b. g. Using values from the walls specified above, solve the system of equations from 6) using the Jacobi method. Iterate until you reach a convergence criteria which you select and state in your answer. At each iteration show your working and give the temperature at each node in degrees centigrade to two decimal places. h. You may wish to use the Matlab code supplied in lab session 4 to help with this part. Increase the number of nodes used in your answer. In your answer plot the scaled residuals as you increase the nodes from 4 to 40. How many nodes would you use to convince a customer that you had achieved grid convergence? Justify this choice. Heat Equation – Unsteady State 2) a. Write down the 1D unsteady heat equation after discretisation using a finite difference, backwards time, central space approach. Write down a set of simultaneous equations for one point in discretised time for the first wall (1) using five nodes in space with equal spacing. b. Express the simultaneous equations in the matrix form, AT = b. c. Using values from the wall (1) specified above, solve the system of equations from 2b for the first time node using the Gauss-Seidel approach. Initially take all unknown temperatures as zero. Only perform one iteration. ABEE3037 – 19/20 CFD for the Built Environment Dr James Pinchin Email: [email protected] Page 3 of 4 How much heat can you take? You may wish to use the Matlab code provided in lab session 4 to help with this question. 3) a. The outside temperature varies from 8 to 28 degrees over a day, changing according to the sinusoidal model (below). Calculate and plot the outside temperature over a day (24hrs). Use any tool which you are comfortable with (Excel, Matlab, plot by hand). Provide the plot in your answer sheet. () = ∆ sin (2 ) + ̅ () = The outside temperature at time d in degrees Celsius. ∆ = The maximum change in temperature above and below the mean in degrees Celsius. = The time of day in hours = The length of the day in hours ̅ = The mean outside temperature in degrees Celsius. b. The air gap in wall (2) is expanded to 3 m wide. It can now be treated as a room with perfectly insulated floor and ceiling. The total domain width (L) is now 3 m + 2*10cm = 3.2 m. Draw the new geometry with boundary conditions. c. Which assumption(s) would we be making about the volume of air inside the simulated room if we use the 1D unsteady heat equation? d. The outside temperature is 0oC (at x = 0 and x = L). The temperature of the walls and air in the room start at 18oC. How long does the temperature in the middle of the room take to drop to zero? You will need to design an appropriate time and space mesh to answer this question. Describe how you decided that the node had reached steady state temperature? How many nodes did you use? e. Using the changing outside temperature from above as the boundary condition at the outside of both walls and assuming that the walls and air volume begin at 18oC what is the range of temperatures (minimum, maximum, range) experienced by a point in the middle of the room over 24hrs? Give your answers in oC to 2 decimal places. You will need to design an appropriate time and space mesh to answer this question. f. Describe and justify the choice of simulation setup* you used to answer the above questions (d and e) – which factors did you take into account when making your choices? g. What is the temperature range experienced at the middle of the room for simulated time 24hrs to 48hrs? Why is this different to the first 24hrs? *initial conditions, space node separation, time node separation. 4) a. Select a temperature range at the centre of the room which would keep an occupant comfortable. Justify this choice including references. b. Given the outside temperature variation and initial room temperature in Q3, use a simulation based on the unsteady 1D heat equation to estimate how thick your walls would need to be in order to keep the temperature variation at the centre of the room in the comfortable range in simulation time 48hrs – 72hrs. (Keep the air width constant and increase the total width (L) with increasing wall thickness). Describe your approach to solving this problem. ABEE3037 – 19/20 CFD for the Built Environment Dr James Pinchin Email: [email protected] Page 4 of 4 Mark Scheme Question Submit Marks Available 1 a Five equations (one per node). 3 1 b Three matrices 3 1 c The estimated steady state temperature at each node in degrees Celsius to 2.d.p. 5 1 d State the node spacing & give a set of equations (one per node). 5 1 e Maximum 1 paragraph (100 words). 5 1 f Three matrices 3 1 g Stated convergence criteria. A series of calculations. At each iteration give estimated steady state temperature at each node in degrees Celsius to 2.d.p. 5 1 h A plot of scaled residuals vs node count. Maximum 1 paragraph (100 words). 11 2 a Five equations (one per node). 5 2 b Three matrices 3 2 c A series of calculations. Give the estimated steady state temperature at each node in degrees Celsius to 2.d.p. 5 3 a A plot of outside temperature vs time. 3 3 b A drawing of a 1D geometry. 3 3 c Maximum 1 paragraph (100 words). 5 3 d A time in integer
seconds. Maximum 1 paragraph (100 words). 5 3 e A maximum and minimum temperature at each node in degrees Celsius to 2.d.p. 5 3 f Maximum 2 paragraphs (200 words). 5 3 g A maximum and minimum temperature at a central node in degrees Celsius to 2.d.p. Maximum 1 paragraph (100 words). 5 4 a A maximum and minimum temperature at a central node in degrees Celsius to 2.d.p. Maximum 1 paragraph (100 words). 5 4 b Maximum 3 paragraphs (300 words). 11 Total 100
