Newbie Question - Water Heater on or off?

Newton's cooling law in convection is a restatement of the differential equation given by Fourier's law:

d Q d t = h · A · ( T ( t ) - T env ) = h · A ? T ( t ) {\displaystyle {{\frac {dQ}{dt}}=h\cdot A\cdot (T(t)-T_{\text{env}})=h\cdot A\Delta T(t)\quad }}

where

Q {\displaystyle Q} is the thermal energy in joules

h {\displaystyle h} is the heat transfer coefficient (assumed independent of T here) (W/(m² K))

A {\displaystyle A} is the heat transfer surface area (m²)

T {\displaystyle T} is the temperature of the object's surface and interior (since these are the same in this approximation)

T env {\displaystyle T_{\text{env}}} is the temperature of the environment; i.e. the temperature suitably far from the surface

? T ( t ) = T ( t ) - T env {\displaystyle \Delta T(t)=T(t)-T_{\text{env}}} is the time-dependent thermal gradient between environment and object

I remeber it well !!

Just to put another perspective on this - we tend to turn our water heater on and off as required - not for reasons of economy or efficiency - but to ensure we don't exceed the current draw from the bollard.  We often stay on CLs and CS's and some of them only provide a supply rated at 10 amps.  It's quite easy to inadvertently trip the circuit breaker if the water heater / kettle / battery charger and perhaps the microwave are all switched on at the same time. If this isn't a consideration, then I agree it makes sense to let the thermostat do its job.

Just do whatever takes your fancy, to many variables to pin it down to a definitive answer. Whether you are in or out most of the day, whether you shower in or use facilities, whether you have loads of dishes and wash them in or use facilities, whether you mind waiting 30 minutes for it to warm up or you want instantaneous, whether there's a z in the month and finally whether you should give two hoots anyway.

Myself I'm not interested in efficiency, if on a CL and no facilities then its on permanently, but if on a CC site then its as and when.

Newton's cooling law in convection is a restatement of the differential equation given by Fourier's law:

d Q d t = h · A · ( T ( t ) - T env ) = h · A ? T ( t ) {\displaystyle {{\frac {dQ}{dt}}=h\cdot A\cdot (T(t)-T_{\text{env}})=h\cdot A\Delta T(t)\quad }}

where

Q {\displaystyle Q} is the thermal energy in joules

h {\displaystyle h} is the heat transfer coefficient (assumed independent of T here) (W/(m² K))

A {\displaystyle A} is the heat transfer surface area (m²)

T {\displaystyle T} is the temperature of the object's surface and interior (since these are the same in this approximation)

T env {\displaystyle T_{\text{env}}} is the temperature of the environment; i.e. the temperature suitably far from the surface

? T ( t ) = T ( t ) - T env {\displaystyle \Delta T(t)=T(t)-T_{\text{env}}} is the time-dependent thermal gradient between environment and object

I remeber it well !!

If only RRM had put it in these simplified terms it would have made soooooo much sense soooooo much sooner. Thanks for clearing it up HD, I've always had a soft spot for Fourier

I hate to cool off the apparent fantastic increased interest in physics on CT but there is a huge error in the first line of that equation.

it should be dQ/dt not dQdt (ie written as a fraction - even though of course differentials are not fractions but can be treated as such in their solving)

Well spotted, Corners.

Yes well spotted, I was wondering how long before the error was noticed 

thank you both - it's a service I provide

Bummer, I never did like that Fourier

It took me hours to work out that formula !!