The gardener built a symmetrical polycarbonate greenhouse on his suburban plot (see Fig. 1). To extend the season, he uses a water tank with a volume of $V = 300~\text{l}$. . During the day, the water is heated by the sun, and at night it cools down, releasing heat to the greenhouse. However, during severe frosts, this heat accumulator is insufficient, so the gardener plans to install an electric heater in the form of a resistor – a long nichrome wire. The heater will be powered by a constant voltage source $U$.
Below is a table with the system parameters that you can consider known throughout the problem.
Parameter Value Density of water $\rho_{\text{w}} = 1000~ \frac{\text{kg}}{\text{m}^3}$ Specific heat capacity of water $c_{\text{w}} = 4200~\frac{\text{Дж}}{\text{kg} \cdot ^\circ \text{C}}$ Length of greenhouse $L = 6.0~\text{m}$ Width of greenhouse $W = 3.0~\text{m}$ Height of greenhouse $H = 2.5~\text{m}$ The length of the roof slope $l = 5.0~\text{m}$ Heat transfer coefficient of polycarbonate $k = 0.35~\frac{\text{W}}{\text{m}^2 \cdot ^\circ \text{C}}$ Minimum allowable temperature inside the greenhouse $T_{\text{min}} = 5~^\circ\text{C}$ Voluem of tank $V = 300~\text{l}$ Duration of night $t_\text{N} = 10~\text{h}$
The heat transfer coefficient is the proportionality factor in the Newton's law of cooling:
$P= kS \Delta T,$
where $P$ is the heat transfer power, $\Delta T$ is the temperature difference, and $S$ is the contact area.
In all parts of the problem, it is necessary to find the numerical value of the required quantity, as well as to express the formula for it using only the given author's notations, unless otherwise stated.
In all parts of this problem, neglect heat loss due to radiation, as well as the heat capacity of the air!
In this part, assume that the outside air temperature at night is constant and equal to $T_\text{out} = -10~^\circ \text{C}$, and the temperature to which the greenhouse (including the water) heats up during the day is $T_\text{in} = 15~^\circ \text{C}$.
In all subsequent parts, you may use the value $S$ in formulaic answers without substituting the expression for it from part A1. In all further parts, the greenhouse floor also remains impermeable to heat!
In question A3, you obtained a lower estimate for the true amount of heat lost by the greenhouse, since the temperature inside the greenhouse $T_\text{inside} \ge T_\text{min} = 5~^\circ \text{C}$.
In the following parts of the problem, you may consider the following parameters as known:
Parameter Value Voltage of the battery $U = 250~\text{V}$ Resistivity of nichrome wire at $0~^\circ \text{С}$ $\rho_0 = 1.1 \cdot 10^{-6}~ \Omega \cdot \text{m}$ Length of wire $\lambda = 150~\text{m}$ Radius of wire $r = 1.0~\text{mm}$ Temperature coefficient of resistance of nichrome $\alpha = 2 \cdot 10^{-4}~^\circ \text{C}^{-1}$ Heat transfer coefficient of nichrome $k_\text{N} = 3.0~\frac{\text{W}}{\text{m}^2 \cdot ^\circ \text{C}} $
Temperature coefficient of resistance $\alpha$ is a coefficient which shows how resistivity changes with temperature:
$$\rho(T) = \rho_0 (1+\alpha \cdot T)$$.
In questions B1-B2, assume the air temperature inside the greenhouse is constant and equal to $T_\text{in} = 20~^\circ \text{C}$.
The gardener is concerned that with a sharp change in $T_\text{out}$, the steady-state temperature inside the greenhouse $T_\text{in}$ will also change sharply, which negatively affects the vegetation in the greenhouse. The gardener has devised a way to avoid this problem: he connects an automated variable resistor in series with the nichrome wire, which can change its resistance when the external temperature changes so that the temperature inside the greenhouse is kept constant. This variable resistor is placed outside, so it does not heat the contents of the greenhouse.
In all the following parts, the air temperature inside the greenhouse is $T_\text{in} = 20~^\circ \text{C} = \text{const}$!
Using the method from part B, it is possible to regulate the amount of heat received by the greenhouse; however, extra power is wasted on useless heating of the variable resistor, causing the gardener to overpay an amount $M$ daily.