AHT Homework 06 AHT 2019 PDF

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Download AHT Homework 06 AHT 2019 PDF

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MECE E6313 Advanced Heat Transfer Arvind Narayanaswamy, Department of Mechanical Engineering, Columbia University. Due date: May 3, 2019. Submit as one pdf file. Share your Google Colab code via Google.

Problem 1 – Derive the Blasius equation f f ′′ + 2f ′′′ = 0 starting from the Navier-Stokes equation under the boundary layer approximation. Determine the boundary conditions on f. Use Google Colab and code snippets from https://github.com/arvindnswamy/ BoundaryLayers to write simple Python codes to solve nonlinear ordinary differential equations. The equation is quite simple: f f ′′ +2f ′′′ = 0. In boundary layer flows, you’ll r see (later, u U∞ in class) that , where = f ′ (η). For fluid at constant velocity over a flat plate, η = y νx U∞ x is the coordinate along the length of the plate, and y perpendicular to it (see figure below). The velocity Uu∞ is a function of x and y, but Prandtl’s showed that the velocity is really r y U∞ a function of √ . We use instead the variable η = y because it is non-dimensional. νx x y Keep in mind that √ is dimensional - painful to deal with. When you go through with x “simplifying” Navier-Stokes equation using the boundary layer assumption, you arrive at the equation ff ′′ + 2f ′′′ = 0. The boundary conditions are: f = 0 at η = 0, f ′ = 0 at η = 0, f ′ → 1 as η → ∞. Here, the superscript ′ means differentiation wrt η. Write a numerical program to solve the above non-linear equation (with the associated boundary conditions). Follow the steps below: 1. Write the entire code and find the following: (a) the value of f”(0) that yields the right BCs as η → ∞.

(b) value of η at which f ′ attains a value of 0.99. This is related to the boundary layer thhickness. Once you know how to solve this problem, you can solve a whole host of non-linear odes with one independent variable (that is the definition of an ode, anyway).

FIG. 1: Flow over a flat plate

2 Problem 2 Starting from the energy equation for flow over a flat plate at constant Pr ′ temperature (in the boundary layer approximation), derive the ODE θ ′′ + fθ = 0. 2 Determine the boundary conditions for θ. Find the relation between the Nusselt number (or heat transfer coefficient) and θ. Use Google Colab to solve the above ODE for θ and N ux for different values of P r. You should choose P r as np.logspace(-4,2,20) to get 20 values of P r distributed equally (on a log scale) between 10−4 to 102 . You can change 20 −1/2 vs P r on a log-log graph to obtain a to 30 or 40 if you want a finer result. Plot N ux Rex result like the one shown in Fig. 2 (that figure is for constant wall flux). From this graph, find the values of C and n in the correlation N ux = CRe1/2 P r n in the limit P r ≪ 1 and P r ≫ 1. Find analytical results for N ux of the form N ux = CRe1/2 P r n (you’ll have to determine the appropriate values of C and n) in the limits P r ≪ 1 and P r ≫ 1. To do this, you have to determine the appropriate form of f (η) for the cases P r ≪ 1 and P r ≫ 1. Compare the numerical results with the analytical results. Hint: There are two ways to solve this problem. The first way is to use the function f as derived from the previous problem. The ODE for θ can be written as follows: #  " y[1]  d y[0] = Pr (1) − f (η )y[1] dη y[1] 2 Of course, in this formulation f (η) (in the Python codes I gave you, t plays the role of η) is treated as a continuous function, like f (η) = η or η 2 , or something like that. However, f (η ) is none of these simple functions. How do you take the result from problem 1 and convert it into a function that you can use? You can create an approximate function for f as follows: ( f ′′(0)η as η → 0 ′ f (η) → (2) 1 as η → ∞ I can create an approximate function as follows:  n  n n  1 ηf ′′ (0) 1 1 ′ ⇒ f (η) = + = 1/n 1 ηf ′′ (0) f ′ (η) [1 + (ηf ′′ (0))n ]

(3)

Any n > 0 is a reasonable choice - n = 1 or 2 are particularly useful for our case because they can be integrated easily to yield analytical expressions for approximations to f (η). Use Worlframalpha to integrate and obtain an analytical expression for f (eta). The only constant you need to supply is f ′′ (0), which you have calculated in problem 1. The second approach is more correct, in the sense that you will not have to approximate for f (η). In this problem, you will solve the momentum and energy problems simultaneously as follows: 2f ′′′(η) + ff ′′(η) =0 Pr ′ θ ′′(η) + fθ =0 2

(4)

These equations can be converted into a form suitable for numerical solution with Python

3 as follows. We’ll define the vector y as follows:     y[0] f  y[1]   f ′       y[2]   f ′′   =   y[3]   θ  y[4] θ′

(5)

Eq. 4 can now be written as:



  y[0] y[1]  y[1]   y[2]   d   y[2]  =  −0.5y[0]y [2]   dη   y[3]   y[4] y[4] − P2r y [4]y [0]

     

(6)

You can solve for θ (or y[3]) directly this way. You might be able to use this technique for Problem 3 also. Problem 3 Find a correlation of the form N ux = CRe1/2 P r n (you’ll have to determine the appropriate values of C and n) for flow over a flat plate with constant wall flux qw by the following methods: ∂T 1 ∂ 2T ∂T . What are the associated ν +v = ∂x ∂y P r ∂y 2 boundary conditions? Keep in mind that u and v are independent of the thermal boundary condition at the plate. An addendum: Try the following substitution T − qw T∞ = − x1/2 θ(η). You have to use this definition of θ to convert the PDE into the k p Pr ′ following equation: θ ′′ + (θ f − f ′ θ ) = 0, where η = y/ νx/U∞ . The solution 2 to this equation is not as easy as that of the ODE for the constant wall temperature case. You can solve it numerically, as you did for the flow problem. The boundary conditions for θ are θ ′ (0) = 1 and θ → 0 as η → ∞. It is possible you can obtain approximate analytical results for the P r ≪ 1 and P r ≫ 1. Naturally, most textbooks do not discuss this problem. If you are really interested in the solution to this problem, your best bet is to look for solutions in journal papers. Sparrow and Lin (International Journal of Heat and Mass Transfer, Vol. 8, pp. 437-448 (1965) is a good source. Max Jakob’s book on heat transfer, available through Hathi Digital Library, is another example. The limitation is that Jakob’s book deals only with the case of an insulated Pr ′ plate. Further details:θ ′′ + (θ f − f ′ θ ) = 0 can be written as: 2     d θ1 θ2 (7) = Pr (θ1 f ′ − θ2 f ) dη θ2 2

1. Exact method, i.e, by solution of u

The above equation can be solved by using Runge Kutta method, just like you did in the previous homework with the Blasius solution. How should be approximate f and f ′ which is required for using the ODE solver (in Python or otherwise). There is a simple way to do the approximation. Consider

4

FIG. 2: Here is what you will/should obtain if you perform an analysis over different values of P r . −1/2 If you plot N ux Re x versus P r , you should get P r 1/3 and P r 1/2 in the P r ≫ 1 and P r ≪ 1 limits. From this graph, you should be able to get the C and m required of your homework.

f ′ (η). This is a function that behaves as f (η) ≈ f ′′ (0)η as η → 0 and f (η) ≈ 1 when 1 1 1 + . If you = ′′ η ≫ 1. Then, a possible approximation is given by ′ f (0)η 1 fapprox ′ ′ plot this fapprox, you’ll see that it deviates quite badly from f (η). You could also try n n  n   1 1 1 , where n is any real number. approximations like + = ′ ′′ fapprox 1 f (0)η f ′′ (0)η ′ . Similarly, f is given by f ′′(0)η 2 /2 as η → 0 and Then, fapprox = 1/n ′′ n [1 + (f (0)η) ] f ′′ (0)η 2 /2 . I have found that n = 4 yields η as η → ∞. Hence, fapprox = [1 + (f ′′ (0)η/2)n ]1/n reasonable approximations for f (η) as well as f ′ (η). 2. Integral method. You’ll have to find the approximate velocity and temperature profiles. The fluid velocity profile will be identical to that we determined for constant wall temperature case. The non-dimensional temperature profiles will change because the thermal boundary condition at y = 0 changes. For both methods, find the expressions for N ux for P r ≪ 1 and P r ≫ 1. Keep in mind that qw Tw now varies with x and the local heat transfer coefficient is defined as h(x) = . Tw (x) − T∞ When Tw was a constant, we strived to find an expression for qw (x). Now, when qw is a constant, |Tw (x) − T∞ | has to increase in order t make up for decreased heat transfer coefficient as x increases....