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Jingying C.
UCSD student Enthusiastic to Share Knowledge
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Physics (Thermodynamics)
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Question:

The question is given: "For the desuperheater, liquid water at state 1 is injected into a stream of superheated vapor entering at state 2. As a result, saturated vapor exits at state 3... Ignoring stray heat transfer and kinetic and potential energy effects, determine the mass flow rate of the incoming superheated vapor, in kg/min." And the steady state data at each state is given by: T1=20 C, p1=0.3 MPa, m'1=6.37 kg/min T2=200 C, p2=0.3 MPa p3=0.3 MPa, Saturated vapor I know that by conservation of mass, the ​mass flow rate is conserved m'1+m'2=m'3 I don't know what data I need to calculate the answer, especially when I'm​ only given one mass flow rate in the setup.

Jingying C.
Answer:

When you don't know what to find, the safe thing to do it to find them all. I know this sounds a lot, but the equation would suggest what variables exist; once you know the variables, try to find all of those values at every state. For the steady state, the energy rate balance equation for a heat exchanger is $$0=\dot{Q_{cv}}+\sum_{i}^{}\dot{m_i}h_i-\sum_{e}^{}\dot{m_e}h_e$$ Since heat transfer is ignored, the equation simplifies to $$0=\dot{m_1}h_1+\dot{m_2}h_2-\dot{m_3}h_3$$ Which suggests that we need $$h$$ at each state. Now you have two equations with two unknowns, where you can solve for $$\dot{m_3}$$.

MATLAB
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Question:

Write Matlab codes to approximate $$\int_{-1}^{1} x e^x dx$$ using the Simpson’s rule. Obtain the approximation for $$n=10$$ steps. I don't know how to write the code for Simpson's rule because of how it alternates between 4 and 2.

Jingying C.
Answer:

The Simpson's rule is given by $$\int_{a}^{b} f(x)d \approx \frac{h}{3}(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+...+4f(x_{n-1})+f(x_n))$$ Note that the summation in the parenthesis is the sum of the sets $$f(x_0)+4f(x_1)+f(x_2)$$ $$f(x_2)+4f(x_3)+f(x_4)$$ $$f(x_4)+4f(x_5)+f(x_6)$$ etc... So I need the index under the $$4f(x)$$ be odd numbers and the index in $$f(x)$$ be even numbers. First creates a function for Simpson's rule in Matlab platform and initializes some variables: $$function [I]=simpson(x0,xn,n)$$ where the variables are x0 - lower bound of the integration interval. xn - upper bound of the integration interval. n - number of iterations. I - result of integral approximation using the Simpson's rule. Initialization: $$h=(xn-x0)/n; $$ is the step size $$I=0;$$ to initialize a result variable $$xv=x0:h:xn;$$ is the x value vectors Review that 2 times any number, $$2n$$, gives an even number and $$2n-1$$ gives an odd number, I will use these expressions to call the x values in the vector xv. Be careful that Matlab index starts from 1 (one step forward from our own index), so xv(1) is actually $$x_0$$, therefore I need the index under the $$4f(x)$$ be even numbers instead and vice versa. Iteration: $$for \space i=1:n/2$$ (n is divide by 2 because it will be multiply by 2 in the iteration) $$\space\space\space\space I=I+h/3*(func(xv(2*i-1))+4*func(xv(2*i))+func(xv(2*i+1)));$$ $$end \space \%for \space loop$$ $$end \space \%function$$ In the iteration, $$func$$ is the external function or subfunction that outputs $$f(x)$$ as specified by the problem; here it is $$x e^x$$ and I will leave that for you to write. We can check the iteration. when i=1, $$2i-1=1, 2i=2, 2i+1=3$$; take one step back to return to our own index, $$func$$ evaluates $$f(x_0)+4f(x_1)+f(x_2)$$. when i=2, $$2i-1=3, 2i=4, 2i+1=5$$; so $$func$$ evaluates $$f(x_2)+4f(x_3)+f(x_4)$$. when i=10/2=5, $$2i-1=9, 2i=10, 2i+1=11$$; so $$func$$ evaluates $$f(x_8)+4f(x_9)+f(x_{10})$$.

Partial Differential Equations
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Question:

Given that the heat distribution inside a uniform sphere of radius $$a$$ can be modeled by $$\frac{\partial u}{\partial t}=\frac{1}{r^2} \frac{\partial}{\partial r}(r^2\frac{\partial u}{\partial r})$$ in the ​spherical coordinate, how do I apply separation of variables to find $$u$$?

Jingying C.
Answer:

To apply the separation of variables method, assume that $$u(r,t)$$ is the multiple of two independent functions: $$u(r,t)=g(r) h(t)$$. Substitute $$g$$ and $$h$$ into the PDE. $$\frac{\partial (gh)}{\partial t}=\frac{1}{r^2} \frac{\partial}{\partial r}(r^2\frac{\partial (gh)}{\partial r})$$ Because $$g$$ is a function of r only, partial derivative $$\frac{\partial}{\partial t}$$ treats $$g$$ as a constant; similar reason applies to $$\frac{\partial}{\partial r}$$ where $$h$$ will be the "constant". So $$g\frac{\partial (h)}{\partial t}=\frac{1}{r^2} \frac{\partial}{\partial r}(r^2h\frac{\partial (g)}{\partial r})$$ I can move the $$h$$ on the right side out of the derivative operator since it's only a function of $$t$$. Partial derivatives will now be ordinary derivative since the function inside the derivative is a function of only the differential variable. $$g\frac{dh}{d t}=\frac{h}{r^2} \frac{d}{d r}(r^2\frac{dg}{d r})$$ DIvide the above equation by $$gh$$ $$\frac{1}{h}\frac{dh}{d t}=\frac{1}{gr^2} \frac{d}{d r}(r^2\frac{dg}{d r})$$ We can see that the left side is a function of the variable $$t$$ only, and the right side is a function of r only. Two functions of different variables can equal to each other only when they both equal to a constant; we called that a "separation constant". $$\frac{1}{h}\frac{dh}{d t}=\frac{1}{gr^2} \frac{d}{d r}(r^2\frac{dg}{d r})=-\lambda$$ This gives two ODEs $$\frac{1}{h}\frac{dh}{d t}=-\lambda$$ and $$\frac{1}{gr^2} \frac{d}{d r}(r^2\frac{dg}{d r})=-\lambda$$ Note that the $$-$$ in front of $$\lambda$$ is not necessary, but it can turn the expression of g into an eigenfunction and simplify the solving process. Lastly, simplify the two expressions $$\frac{dh}{d t}+\lambda h=0$$ and $$\frac{d}{d r}(r^2\frac{dg}{d r})+\lambda gr^2=0$$

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