$(10$ pts$)$ In the idealized case, where fouling and heat conduction in the device material between the streams are neglected, the overall convection coefficient of a heat exchanger can be expressed as

$U_m=\frac{1}{\frac{1}{x}+\frac{1}{y}},$

where $x$ and $y$ are the convection coefficients of the individual streams. Suppose you are faced with the proposition of designing an exchanger in which you could adjust $x$ and $y,$ but only in a way that they vary inversely, $x*y=C,$ where $C$ is a constant. For instance, if you doubled $x,$ then $y$ would be halved and vice$-$versa. If the best design maximizes $U_m,$ show that it is obtained when $x=y,$ i$.$e$.$ when the convection in the two streams is "impedance matched". Note that $x,y,$ and therefore $C,$ as well, are all positive numbers.$(10$ pts$)$ In the idealized case, where fouling and heat conduction in the device material between the streams are neglected, the overall convection coefficient of a heat exchanger can be expressed as

$U_m=\frac{1}{\frac{1}{x}+\frac{1}{y}},$

where $x$ and $y$ are the convection coefficients of the individual streams. Suppose you are faced with the proposition of designing an exchanger in which you could adjust $x$ and $y,$ but only in a way that they vary inversely, $x*y=C,$ where $C$ is a constant. For instance, if you doubled $x,$ then $y$ would be halved and vice$-$versa. If the best design maximizes $U_m,$ show that it is obtained when $x=y,$ i$.$e$.$ when the convection in the two streams is "impedance matched". Note that $x,y,$ and therefore $C,$ as well, are all positive numbers.